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Uniform Convergence
APPENDIX A UNIFORM C O N V ER GE NCE
We saw in Chapter 2 that Cauchy stated the first modem definition of the convergence of series in his Cours d'analyse de l'Ecole Royale Polytechnique of 1821. The working version of this definition today was given by Heine in 1872, and in order to reproduce it in a form that will serve our purposes we shall accept the following notation. S will denote a set of points in the real line, in the plane, or in space. A point of S will be denoted by s except when we specifically mean a real number, in which case it will be denoted by x. Definition A. 1 For eachpositive integer n, let fn : S ~ The series
IR be a real-valued function.
oo
n=l
is said to be convergent or to converge to a function f : S ~ IR, called its sum, if and only if f o r every s in S and any E > 0 there is a positive number N = N (E, s) such that ~
f k ( s ) -- f (s) <
k=l
f o r every n > N.
The notation N ( ~ , s ) is meant to emphasize the fact that this number generally depends on the choice of both E and s. The same definition can be used for the convergence of a sequence {fn } rather than a series of functions. One simply replaces Y']~=l fk with fn. After stating his definition and proving a number of convergence tests, Cauchy then stated an incorrect theorem: the sum of a convergent series of continuous functions is continuous. It was the Norwegian mathematician Niels Henrik Abel (1802-1829) who first called attention to this error by pointing out that the sum of the Fourier series of f (x) = x, whose terms are continuous functions, is discontinuous at each x = (2n + 1)zr, where n is any integer (see Example 2.9 of Chapter 2). Then, in 1829, Dirichlet's theorem on the convergence of Fourier series made this abundantly clear. This is not mentioned 506
UNIFORM CONVERGENCE
507
to show a blemish in Cauchy's work, but because of its connection with an important discovery. But before that, let us place this matter in a more general context. We are interested in determining whether or not the properties of continuity, differentiability, or integrability of the f n carry over to the sum of the series. Moreover, for functions with domain in the real line, we want to know if and under what conditions differentiation and integration can be carried out term by term; that is, whether or not the expressions fn
=
fn~
n=l
and
n=l
/ab
b L =
n=l
fn n=l
are valid. In general, the answer to each of our questions is no, as shown by the following very simple example. E x a m p l e A.1 The series OO
EX(1 n=l
--X2) n
converges in the interval [ - 1 , 1] to the function 0 f(x)
-
if
x-O,
+1
if
x#O,
4-1.
1
--x x
This is clear for x = 0, 4-1 because then every term of the series vanishes. Otherwise, put r = 1 - x 2 and observe that (r + r 2 + - . .
+ rn)(1
-- r) = r -- r n+l
= r(1 --
rn).
Then ~-~ x(1 k=l
X2) k
1 - x 2 = x(1 -- x2)[1 -- (1 -- x2) n] 1 - - ( 1 - - x 2) X
1 --X 2
(1 - - X 2 ) n+l
x
and, since 0 < 1 - x 2 < 1, the right-hand side can be made smaller than any given E > 0 by simply choosing n large enough. Therefore, the series converges, but, while every term is continuous, differentiable, and integrable on [ - 1 , 1], its sum is not. Probably at Dirichlet's prompting, one of his students, Phillip Ludwig von Seidel (1821-1896), was led to investigate this situation. Consider, for instance, a series of continuous functions that converges to a function discontinuous a t a point so. About 1848, Seidel and, independently, Sir George Gabriel Stokes (1819-1903) made the observation that, if E > 0 is given, no N can be chosen so that the inequality in Definition A. 1
508
APPENDIX A
is satisfied for all s near so. More precisely, for each such s there is an N such that the inequality in Definition A. 1 holds, but the same N does not work f o r all s, for, the closer s is to so, the larger N has to be. Seidel and Stokes failed to pursue the matter, but it seems clear that, if we include in the definition of convergence the stipulation that N be independent of s, then we have a new, more restrictive kind of convergence that, as we shall see, makes it impossible for the sum of the series to be discontinuous. Definition A.2 Let S and the fn be as in Definition A. 1. Then the series Y~n~=l fn converges u n i f o r m l y on S to a sum f if and only if f o r any ~ > 0 there is a positive number N = N(E) such that
~-~ fk(S) -- f (s)
k=l
f o r every n > N and all s in S. The series in Example A.1 is not uniformly convergent. Indeed, if 0 < ~ < 1 is given, if we assume that N can be chosen to satisfy Definition A.2, and if x > 0 is chosen to be so small that 0 < 2 N+2 x < E, then x < 1/2 so that 1 - x 2 > 1/2. If we choose n -- N + 1, we have (1 - x 2 ) n+l
1
x
2N+2x
> E,
a contradiction. As it happens, the idea of a different kind of convergence was not entirely new. Already in 1838 Christof Gudermann (1798-1852) had referred to a kind of convergence at the same rate that is the precursor of uniform convergence. But the importance of the concept escaped him, as it would escape Seidel and Stokes later on. This realization was left to Gudermann's student Karl Theodor Wilhelm Weierstrass (1815-1897), one of the giants of modern mathematics. Not particularly inspired by the lectures at the University of Bonn, where he was a student, in 1839 he went to Miinster to attend Gudermann's lectures, attracted by some notes from these that he had had the opportunity to see at Bonn. Gudermann was to influence Weierstrass' research on analytic functions, and it is quite likely that, while at Mfinster, they discussed the new concept of convergence. Weierstrass never finished his doctorate and became a Gymnasium teacher in 1841, a position that he held until 1854. During this time he produced an incredible amount of first-rate research in manuscript form but, unconcerned with questions of priority, it remained unpublished for the longest time. The fact that he referred to uniform convergencem gleichmiissige Convergenz--in an 1841 manuscript supports the idea that he may have learned about it from Gudermann. Weierstrass' many research achievements eventually earned him a position at the University of Berlin in 1856, where he frequently discussed uniform convergence. He defined it formally, for functions of several variables, in a paper published in 1880. As in Cauchy's case regarding ordinary convergence, the importance of Weierstrass' contribution stems from the fact that he realized the usefulness of uniform
UNIFORM CONVERGENCE 509
KARL WEIERSTRASS Photograph by the author from Weierstrass' Mathematische Werke of 1903.
convergence and incorporated it in theorems on the integrability and differentiability of series of functions term by term. Before we show this, we must point out a shortcoming that Definitions A. 1 and A.2 share. Application of these definitions requires an a priori knowledge of the sum f , but this may not be available. To obtain a result that does not require this knowledge we shall use the well-known Cauchy's criterion for the convergence of sequences of 1821. In spite of its name, this criterion had already been discovered in 1817 by the Bohemian priest Bernardus Placidus Johann Nepomuk Bolzano (1781-1848), a professor of the philosophy of religion at Prague. L e m m a A.1 (Cauchy's Criterion). Let {Xn} be a sequence o f real numbers. I f f o r every ~ > 0 there is a positive integer N such that IXm -- Xn [ < 6
f o r all m and n > N, then {Xn } converges.
Proof. For a given ~ > 0 let N be as stated above. If all but a finite number of the Xn are equal, then they are equal to a certain number a. In particular, there is one
510
APPENDIX A
n > N such that
Xn :
a. Then, if m > N, we have IXm - a l -- IXm - Xn I < E,
which shows that a is the limit of the sequence {Xn }. If infinitely many of the Xn are distinct and if M is an integer larger than N, then the inequality IXM - Xn[ < E for n > N means that all the points Xn with n > N are in the interval [XM -- e, x M + e]. If we split this interval in half using its midpoint, then at least one of the resulting subintervals, call it [a l, bl ], contains infinitely many of the Xn. Splitting [al, bl] in half again, one of the resulting subintervals, call it [a2, b2], contains infinitely many of the Xn. This splitting process can be continued indefinitely, obtaining for each positive integer k a subinterval [ak, bk] that contains infinitely many of the Xn. We shall show that there is exactly one point contained in the intersection of these shrinking intervals and that this point is the limit of {Xn }. One of the defining axioms of the real number system states that, if an infinite set S of real numbers is bounded above, then there is a smallest real number, called the s u p r e m u m of S, that is not smaller than any of those in S. 1 Let a be the supremum of all the ak, a set bounded above by bl. Similarly, let b be the infimum of all the bk, defined as the largest real number that is not larger than any of the bk. It is clear that a < b and that both a and b are contained in [ak, bk] for each k. Since the length of [ak, bk ] approaches zero as k ~ cx~, it follows that a -- b and that this is the only point contained in all the [ak, bk]. Now let k be so large that the length of [ak, bk] is less than e. Since [ak, bk] contains a and infinitely many of the Xn, there is an m > N such that [Xm - a l < E. Using the triangle inequality it follows that, if n > N, [Xn - a l ~ [Xn - X m I -~- IXm - a l < ~ + E : 2~,
and this proves that Xn ~
a as n --+ cx~, Q.E.D.
It was Bolzano's original and ingenious idea to repeatedly bisect an interval to close down on a single point that must be the limit of the sequence. We can now use this basic fact from the theory of the convergence of sequences to establish some results on the convergence of series.
Theorem A.1 (Cauchy's Criterion for Uniform Convergence). A series o f f u n c -
Y~m~_l f n converges uniformly on a set S if and only if f o r every E > 0 there is a positive n u m b e r N such that tions
~-'~ fk(S) k--m
< E
f o r all n > m > N and all s in S.
1See,for instance, W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976, page 8.
UNIFORM CONVERGENCE
511
Proof. (if) L e m m a A. 1 applied to the sequence of partial sums of the given series shows that it converges for each s in S, and we can define oo
f (s) = Z
fn(S).
n=l
Let e and N be as above and, for brevity, define n
Fn(s)-
Zfk(s). k=l
Since Fn(s) --+ f ( s ) for each s as n --+ oe, we can choose a positive integer m = m(s) > N so large that I F m ( s ) - f ( s ) l < E. If n > N then, for any s in S and with m as above, we have
IFn(s) - f ( s ) l _< IFn(s) - Fm(s)l + [Fm(s) - f ( s ) l < e + e = 2e for all s in S, showing that Fn --+ f uniformly on S. (only if) If Y~n=l oc fn = f uniformly on S and ~ > 0 is given, then
]Fn(s)- f(s)] < 2 for n large enough and all s in S. Thus, if n >_ m,
~--~ fn(s) k=m
= ]Fn(s) - Fm-l(S)[ < ]Fn(s) - f ( s ) l + I f ( s ) - Fm-l(S)] < E
for m and n large enough and all s in S, Q.E.D. The next theorem gives only a sufficient condition for uniform convergence, but it is quite useful in many applications. T h e o r e m A.2 (The W e i e r s t r a s s M-test). Let y~meC:l fn be a series of functions defined on a set S, and suppose that f o r each n there is a number Mn such that
Ifn(s)l <_ Mn f o r all s in S. If the series y-~n~=l Mn converges, then Y~n=l fn converges uniformly Oil S. Proof. Given ~ > 0 and applying L e m m a A. 1 to the sequence of partial sums of the given series, there is an N such that
fk(s) k=m
< ~Mk
for n _> m > N and all s in S. The result follows from Theorem A. 1, Q.E.D.
512
APPENDIXA
In a way, the Weierstrass M-test represents a case of overkill because, not only does it show that Y~n~176fn converges uniformly, but also it shows that Y'~n~=l Ifn I converges uniformly. However, ~n~=l fn may in some cases converge uniformly while ~nCX~=lIf.I does not converge even at a single point (Exercise A. 16). Even if Y~n=l fn converges uniformly and )--~nC~_lIfnl converges, this last series need not converge uniformly (Exercise A. 15). More refined convergence tests can be obtained, for series of a special form, using a formula due to Abel known as summation by parts. For each positive integer n let f n " S ~ I R and gn 9 S --+ I R be functions and define Fn = Y ' ~nk = l fk. If n > m > 1, we have fkgk = ~ k=m
(Fk -- Fk-1)gk
k=m n-1 k=m
k=m-1 n-1
-- Fngn - Fm-1 gm - ~
Fk(gk+l -- gk).
k=m
Using this formula--whose name refers to the fact that its general structure is similar to the formula for integration by parts--Abel gave a convergence test for numerical series, whose generalization for series of functions is still associated with his name. Theorem A.3 (Abel's Test for Uniform Convergence). For each positive integer n, let fn : S ~ IR and gn : S ~ IR be functions such that (i) ~ff-'~nC~_lf n converges uniformly on S, (ii) either gn(s) < gn+l(S) or gn+l(S) < gn(s) for each n and all s in S, and (iii) there is a constant M > 0 such that Ign (s)l < M for each n and all s in S. Then the series oo
Y ~ fngn
n=l converges uniformly on S.
Proof. Let f be the sum of En~176f n . Since, for n > m > 1,
n-1 Y ~ (gk+l
--
gk) = gn -- gm,
k=m
the summation by parts formula gives fkgk = Fngn
Fm-lgm
k=m
~
F t ( g k + l - gk)
f
n
gm
k=m
Y~(gk+l k=m
n-1 = (Fn -
f)gn
-
(Fm-1
-
f)gm
-
Y~ k=m
(Fk
-
f)(gk+l-
gk).
gk)
]
UNIFORM CONVERGENCE
513
By Definition A.2, given E > 0 there is an N such that IFk(s) -- f ( s ) l < E / 4 M for k > N and all s in S. Then, if n > m > N, we have k=m fkgk
< ~--~(Ignl + Igml + Ign -- gml) < ~--~2(Ignl + Igml) < E
on S. The result follows from Theorem A. 1, Q.E.D. The applicability of Abel's test is limited because we have to use another test first to determine the uniform convergence of ~-~n%l f n . This cannot be the Weierstrass M-test, (x) o~ because, if this test applies to ~ n = 1 fn, then it also applies to Y~n=l fn gn in view of the condition Ign(t)l < M. Abel's test is very useful, however, when the fn are constants rather than functions. We have the following corollary. C o r o l l a r y If Y~n=l an is a convergent series of numbers and if gn" S ---> IR are functions that satisfy conditions (ii) and (iii) of the theorem, then the series oo
~-~ an gn n=l converges uniformly on S. We are finally ready to state conditions under which continuity, integrability, and differentiability of the terms of a series of functions carry over to its sum. The first result is due to Abel who essentially proved it in 1826 but let the underlying concept of uniform convergence escape unnoticed. T h e o r e m A.4 The sum of a uniformly covergent series of continuous functions on a set S is continuous on S. P r o o f . If ~-~n= c~ 1 fn = f uniformly on S, if each fn is continuous on S, if Fn denotes the nth partial sum of the series, and if s and so are in S, then
I f ( s ) - f(s0)l < I f ( s ) - Fn(s)l + IFn(s) - Fn(sO)l + IFn(sO) - f(s0)l. By the uniform convergence of the series, the first and third terms on the fight can be made arbitrarily small by choosing n large enough. The second term can be made arbitrarily small by choosing s close enough to so. Then the left-hand side can also be made arbitrarily small by choosing s close enough to so, Q.E.D. The two remaining results are due to Weierstrass. The set S is now an interval in IR. T h e o r e m A.5 /f ~-~n%l fn is a uniformly convergent series o f integrable functions defined on an interval [a, b], then A = n=l
A n=l
that is, the sum of the series is integrable and may be integrated term by term.
514
APPENDIX A
Sn
Proof. If Fn = }--]~=l fk and if f denotes the sum of the series, let be the supremum of the set {If(x) - Fn(x)l " a < x _< b} (this concept was introduced in the proof of L e m m a A. 1). If a = x0 < xl < . . . < Xn = b is a partition of [a, b], let Mi andmi be the supremum and infimum of f on each (xi-1, xi), let Im be the supremum of the s u m s y]4n__l corresponding to f for all partitions of [a, b], and let IM n be the infimum of the sums ~ i = 1 Mi(xi - xi-1). We have
mi(xi - xi-1)
/a
(Fn - s~) <_ Im <_ l u <_
fa
(Fn + Sn),
and then 0 < IM -- Im <_ 2Sn (b - a ) . Since Sn --~ 0 as n ~ e~ because the convergence is uniform, we conclude that Im = IM. By Definition 2.2, this means that f is integrable. Finally,
f-
Fn =
( f - Fn) <_
I f - Fnl < s n ( b - a ) ,
which approaches zero as n --~ ec, Q.E.D.
Theorem A.6 /f Y]~n=l fn converges on an interval [a, b], if each fn has a continuous derivative on [a , b], and if the series ~--~n=l ~ fn' converges uniformly on [a, b], then ~-'~n~-_l fn is differentiable on [a, b] and = n=l
f~" n=l
that is, the series can be differentiated term by term. Proof. Applying Theorem A.5 to the series of derivatives, which are integrable because they are continuous, and for each x in [a, b] we obtain
faX~ n=l
=
~fa
n=l
fn' =
~-~[fn(X)
f~(a)]
n=l
=
f~(a). ~-~f n ( x ) -- n~-~ =l
n=l
Then, differentiating with respect to the upper limit of integration,
Zfn: n=l
(5)'fn n=l
on [a, b], Q.E.D. The hypotheses in Theorem A. 6 are stronger than necessary. This was done for the sake of giving a simple proof, but the conclusion is valid if the fn are assumed only to be differentiable on [a, b] and if the series Y]~n=l fn converges at a single point. 1 Note also that, in contrast with Theorems A.4 and A.5, it is the uniform convergence of Y']~n=l oo fn, that is required rather than that of Y'~n=l oo fn. The uniform convergence of Y]~n~-_l fn would not suffice, as illustrated by the following example. 1See, for instance, W. Rudin, op. cit., page 152.
EXERCISES 515
E x a m p l e A.2 The series
n:~ --~~ ~-~ sin n 3x converges uniformly on any interval by the Weierstrass M-test because the series cr
1
is known to converge. However, the series of derivatives o<3
Z n cos n3x n=l diverges at every point x - k:r, where k is an integer, because I cos n3krrl = 1, and the nth term of the series does not approach zero as n ~ cr
EXERCISES
A.1
Show that the series 1 ~n2 [sin(n3 + 7t.)(x 2 + y2) + x 3] n-1
converges uniformly on S = { (x, y) ~ IR2" x 2 + y2 < 1 }. A.2
Prove that, if a series of real numbers Y~,,~I an converges absolutely, then o@
o@ IE"''"'qL
an COSnx
and
n=l
A.3
~
converge uniformly on IR. Prove that, if a series of real numbers y-~n~=lan converges absolutely, then the series o@ n x n=l
converges uniformly on [0, ~ ) . A.4
an sin nx
n=l
)n
Show that the series o@
,,=1
converges uniformly on [e, e2].
101ogx
516
APPENDIX A
A.5
Show t h a t ~_,n~176 (xn/n !) converges uniformly on [ - a , a] for any a > 0. Is the convergence uniform on IR?
A.6
Show that the series
(x)
y~ (--1)n e-n2(x2+y2) n
n=l
converges uniformly on IR2. A.7
Show that the series (-1) n
n=~ ~
x
COS--n
converges uniformly on [ - J r / 4 , Jr/4]. oo
n
A.8
Showthat y-]~n=0(x / n ! ) converges to a continuous function f onlR. Showthat f ' = f on IR.
A.9
Show that
n=l ~ sin nx = 2 n=l (2n - 1)4 A.10
Show that for ]xl < 1 ( _ 1) n+l ~ x
log(1 + x ) = ~
n
n
n=l
(Hint:
consider the geometric series ~-]~,,~0( - x ) n ' which is uniformly convergent for Ixl
A.11
Show that the series ~
and
X 2n + 1
n=0 (2n + 1)v9
n--0
(2n)!
can be differentiated term by term and find their derivatives. On what intervals are these derivatives valid? A.12
Prove Dirichlet's test for uniform convergence: for every positive integer n, let fn : S ~ IR and gn : S --~ IR be functions such that n (i) there is a constant M such that IY]~k=l < M for each n and all s in S, (ii) gn+l (s) _< gn(S) for each n and all s in S, and (iii) gn ~ 0 uniformly on S. Then the series ~,,~--1 fngn converges uniformly on S.
A(s)l
A.13
Use Dirichlet's test to prove the uniform convergence of the series ~
1
~-~-n--1
COS
nx
n
on the interval [8, Jr - 8] for any 0 < 8 < zr/2. A.14
Prove the following corollary of Dirichlet's test for uniform convergence. For each positive integer n, let gn" S ~ lR be a function such that (i) gn+l (S) < gn(S) for each n and all s in S and (ii) gn -~ 0 uniformly on S. Then )-~n~__1( - 1)n gn converges uniformly on S.
EXERCISES
517
A.15
Define fn'[O, 1] ~ IR by fn(X) = (--x)n(1 -- X). Show that Z n ~ l fn converges uniformly on [0, 1]. Does y~n~l Ifnl converge on [0, 1]? Does it converge uniformly?
A.16
I f a > 0, define fn'[O,a] --+ IR by f ( x ) = ( - 1 ) n ( x -t-n)/n 2. Show that y~n~=l f,, oo converges uniformly on [0, a]. Does Y~n-I Ifnl converge at any x E IR? Show that the series
A.17
o~ Z(--x)n(1
--
x)e-n2y
n=l
converges uniformly on S = { (x, y) 6 IR2:0 < x < 1, y > 0 } (Hint: use Exercise A.15). A.18
Show that the series n~l =
(-1)n(xwn) k=l
converges uniformly on [0, 1] (Hint: use Exercise A. 16).
We saw in Chapter 2 that Cauchy stated the first modem definition of the convergence of series in his Cours d'analyse de l'Ecole Royale Polytechnique of 1821. The working version of this definition today was given by Heine in 1872, and in order to reproduce it in a form that will serve our purposes we shall accept the following notation. S will denote a set of points in the real line, in the plane, or in space. A point of S will be denoted by s except when we specifically mean a real number, in which case it will be denoted by x. Definition A. 1 For eachpositive integer n, let fn : S ~ The series
IR be a real-valued function.
oo
n=l
is said to be convergent or to converge to a function f : S ~ IR, called its sum, if and only if f o r every s in S and any E > 0 there is a positive number N = N (E, s) such that ~
f k ( s ) -- f (s) <
k=l
f o r every n > N.
The notation N ( ~ , s ) is meant to emphasize the fact that this number generally depends on the choice of both E and s. The same definition can be used for the convergence of a sequence {fn } rather than a series of functions. One simply replaces Y']~=l fk with fn. After stating his definition and proving a number of convergence tests, Cauchy then stated an incorrect theorem: the sum of a convergent series of continuous functions is continuous. It was the Norwegian mathematician Niels Henrik Abel (1802-1829) who first called attention to this error by pointing out that the sum of the Fourier series of f (x) = x, whose terms are continuous functions, is discontinuous at each x = (2n + 1)zr, where n is any integer (see Example 2.9 of Chapter 2). Then, in 1829, Dirichlet's theorem on the convergence of Fourier series made this abundantly clear. This is not mentioned 506
UNIFORM CONVERGENCE
507
to show a blemish in Cauchy's work, but because of its connection with an important discovery. But before that, let us place this matter in a more general context. We are interested in determining whether or not the properties of continuity, differentiability, or integrability of the f n carry over to the sum of the series. Moreover, for functions with domain in the real line, we want to know if and under what conditions differentiation and integration can be carried out term by term; that is, whether or not the expressions fn
=
fn~
n=l
and
n=l
/ab
b L =
n=l
fn n=l
are valid. In general, the answer to each of our questions is no, as shown by the following very simple example. E x a m p l e A.1 The series OO
EX(1 n=l
--X2) n
converges in the interval [ - 1 , 1] to the function 0 f(x)
-
if
x-O,
+1
if
x#O,
4-1.
1
--x x
This is clear for x = 0, 4-1 because then every term of the series vanishes. Otherwise, put r = 1 - x 2 and observe that (r + r 2 + - . .
+ rn)(1
-- r) = r -- r n+l
= r(1 --
rn).
Then ~-~ x(1 k=l
X2) k
1 - x 2 = x(1 -- x2)[1 -- (1 -- x2) n] 1 - - ( 1 - - x 2) X
1 --X 2
(1 - - X 2 ) n+l
x
and, since 0 < 1 - x 2 < 1, the right-hand side can be made smaller than any given E > 0 by simply choosing n large enough. Therefore, the series converges, but, while every term is continuous, differentiable, and integrable on [ - 1 , 1], its sum is not. Probably at Dirichlet's prompting, one of his students, Phillip Ludwig von Seidel (1821-1896), was led to investigate this situation. Consider, for instance, a series of continuous functions that converges to a function discontinuous a t a point so. About 1848, Seidel and, independently, Sir George Gabriel Stokes (1819-1903) made the observation that, if E > 0 is given, no N can be chosen so that the inequality in Definition A. 1
508
APPENDIX A
is satisfied for all s near so. More precisely, for each such s there is an N such that the inequality in Definition A. 1 holds, but the same N does not work f o r all s, for, the closer s is to so, the larger N has to be. Seidel and Stokes failed to pursue the matter, but it seems clear that, if we include in the definition of convergence the stipulation that N be independent of s, then we have a new, more restrictive kind of convergence that, as we shall see, makes it impossible for the sum of the series to be discontinuous. Definition A.2 Let S and the fn be as in Definition A. 1. Then the series Y~n~=l fn converges u n i f o r m l y on S to a sum f if and only if f o r any ~ > 0 there is a positive number N = N(E) such that
~-~ fk(S) -- f (s)
k=l
f o r every n > N and all s in S. The series in Example A.1 is not uniformly convergent. Indeed, if 0 < ~ < 1 is given, if we assume that N can be chosen to satisfy Definition A.2, and if x > 0 is chosen to be so small that 0 < 2 N+2 x < E, then x < 1/2 so that 1 - x 2 > 1/2. If we choose n -- N + 1, we have (1 - x 2 ) n+l
1
x
2N+2x
> E,
a contradiction. As it happens, the idea of a different kind of convergence was not entirely new. Already in 1838 Christof Gudermann (1798-1852) had referred to a kind of convergence at the same rate that is the precursor of uniform convergence. But the importance of the concept escaped him, as it would escape Seidel and Stokes later on. This realization was left to Gudermann's student Karl Theodor Wilhelm Weierstrass (1815-1897), one of the giants of modern mathematics. Not particularly inspired by the lectures at the University of Bonn, where he was a student, in 1839 he went to Miinster to attend Gudermann's lectures, attracted by some notes from these that he had had the opportunity to see at Bonn. Gudermann was to influence Weierstrass' research on analytic functions, and it is quite likely that, while at Mfinster, they discussed the new concept of convergence. Weierstrass never finished his doctorate and became a Gymnasium teacher in 1841, a position that he held until 1854. During this time he produced an incredible amount of first-rate research in manuscript form but, unconcerned with questions of priority, it remained unpublished for the longest time. The fact that he referred to uniform convergencem gleichmiissige Convergenz--in an 1841 manuscript supports the idea that he may have learned about it from Gudermann. Weierstrass' many research achievements eventually earned him a position at the University of Berlin in 1856, where he frequently discussed uniform convergence. He defined it formally, for functions of several variables, in a paper published in 1880. As in Cauchy's case regarding ordinary convergence, the importance of Weierstrass' contribution stems from the fact that he realized the usefulness of uniform
UNIFORM CONVERGENCE 509
KARL WEIERSTRASS Photograph by the author from Weierstrass' Mathematische Werke of 1903.
convergence and incorporated it in theorems on the integrability and differentiability of series of functions term by term. Before we show this, we must point out a shortcoming that Definitions A. 1 and A.2 share. Application of these definitions requires an a priori knowledge of the sum f , but this may not be available. To obtain a result that does not require this knowledge we shall use the well-known Cauchy's criterion for the convergence of sequences of 1821. In spite of its name, this criterion had already been discovered in 1817 by the Bohemian priest Bernardus Placidus Johann Nepomuk Bolzano (1781-1848), a professor of the philosophy of religion at Prague. L e m m a A.1 (Cauchy's Criterion). Let {Xn} be a sequence o f real numbers. I f f o r every ~ > 0 there is a positive integer N such that IXm -- Xn [ < 6
f o r all m and n > N, then {Xn } converges.
Proof. For a given ~ > 0 let N be as stated above. If all but a finite number of the Xn are equal, then they are equal to a certain number a. In particular, there is one
510
APPENDIX A
n > N such that
Xn :
a. Then, if m > N, we have IXm - a l -- IXm - Xn I < E,
which shows that a is the limit of the sequence {Xn }. If infinitely many of the Xn are distinct and if M is an integer larger than N, then the inequality IXM - Xn[ < E for n > N means that all the points Xn with n > N are in the interval [XM -- e, x M + e]. If we split this interval in half using its midpoint, then at least one of the resulting subintervals, call it [a l, bl ], contains infinitely many of the Xn. Splitting [al, bl] in half again, one of the resulting subintervals, call it [a2, b2], contains infinitely many of the Xn. This splitting process can be continued indefinitely, obtaining for each positive integer k a subinterval [ak, bk] that contains infinitely many of the Xn. We shall show that there is exactly one point contained in the intersection of these shrinking intervals and that this point is the limit of {Xn }. One of the defining axioms of the real number system states that, if an infinite set S of real numbers is bounded above, then there is a smallest real number, called the s u p r e m u m of S, that is not smaller than any of those in S. 1 Let a be the supremum of all the ak, a set bounded above by bl. Similarly, let b be the infimum of all the bk, defined as the largest real number that is not larger than any of the bk. It is clear that a < b and that both a and b are contained in [ak, bk] for each k. Since the length of [ak, bk ] approaches zero as k ~ cx~, it follows that a -- b and that this is the only point contained in all the [ak, bk]. Now let k be so large that the length of [ak, bk] is less than e. Since [ak, bk] contains a and infinitely many of the Xn, there is an m > N such that [Xm - a l < E. Using the triangle inequality it follows that, if n > N, [Xn - a l ~ [Xn - X m I -~- IXm - a l < ~ + E : 2~,
and this proves that Xn ~
a as n --+ cx~, Q.E.D.
It was Bolzano's original and ingenious idea to repeatedly bisect an interval to close down on a single point that must be the limit of the sequence. We can now use this basic fact from the theory of the convergence of sequences to establish some results on the convergence of series.
Theorem A.1 (Cauchy's Criterion for Uniform Convergence). A series o f f u n c -
Y~m~_l f n converges uniformly on a set S if and only if f o r every E > 0 there is a positive n u m b e r N such that tions
~-'~ fk(S) k--m
< E
f o r all n > m > N and all s in S.
1See,for instance, W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976, page 8.
UNIFORM CONVERGENCE
511
Proof. (if) L e m m a A. 1 applied to the sequence of partial sums of the given series shows that it converges for each s in S, and we can define oo
f (s) = Z
fn(S).
n=l
Let e and N be as above and, for brevity, define n
Fn(s)-
Zfk(s). k=l
Since Fn(s) --+ f ( s ) for each s as n --+ oe, we can choose a positive integer m = m(s) > N so large that I F m ( s ) - f ( s ) l < E. If n > N then, for any s in S and with m as above, we have
IFn(s) - f ( s ) l _< IFn(s) - Fm(s)l + [Fm(s) - f ( s ) l < e + e = 2e for all s in S, showing that Fn --+ f uniformly on S. (only if) If Y~n=l oc fn = f uniformly on S and ~ > 0 is given, then
]Fn(s)- f(s)] < 2 for n large enough and all s in S. Thus, if n >_ m,
~--~ fn(s) k=m
= ]Fn(s) - Fm-l(S)[ < ]Fn(s) - f ( s ) l + I f ( s ) - Fm-l(S)] < E
for m and n large enough and all s in S, Q.E.D. The next theorem gives only a sufficient condition for uniform convergence, but it is quite useful in many applications. T h e o r e m A.2 (The W e i e r s t r a s s M-test). Let y~meC:l fn be a series of functions defined on a set S, and suppose that f o r each n there is a number Mn such that
Ifn(s)l <_ Mn f o r all s in S. If the series y-~n~=l Mn converges, then Y~n=l fn converges uniformly Oil S. Proof. Given ~ > 0 and applying L e m m a A. 1 to the sequence of partial sums of the given series, there is an N such that
fk(s) k=m
< ~Mk
for n _> m > N and all s in S. The result follows from Theorem A. 1, Q.E.D.
512
APPENDIXA
In a way, the Weierstrass M-test represents a case of overkill because, not only does it show that Y~n~176fn converges uniformly, but also it shows that Y'~n~=l Ifn I converges uniformly. However, ~n~=l fn may in some cases converge uniformly while ~nCX~=lIf.I does not converge even at a single point (Exercise A. 16). Even if Y~n=l fn converges uniformly and )--~nC~_lIfnl converges, this last series need not converge uniformly (Exercise A. 15). More refined convergence tests can be obtained, for series of a special form, using a formula due to Abel known as summation by parts. For each positive integer n let f n " S ~ I R and gn 9 S --+ I R be functions and define Fn = Y ' ~nk = l fk. If n > m > 1, we have fkgk = ~ k=m
(Fk -- Fk-1)gk
k=m n-1 k=m
k=m-1 n-1
-- Fngn - Fm-1 gm - ~
Fk(gk+l -- gk).
k=m
Using this formula--whose name refers to the fact that its general structure is similar to the formula for integration by parts--Abel gave a convergence test for numerical series, whose generalization for series of functions is still associated with his name. Theorem A.3 (Abel's Test for Uniform Convergence). For each positive integer n, let fn : S ~ IR and gn : S ~ IR be functions such that (i) ~ff-'~nC~_lf n converges uniformly on S, (ii) either gn(s) < gn+l(S) or gn+l(S) < gn(s) for each n and all s in S, and (iii) there is a constant M > 0 such that Ign (s)l < M for each n and all s in S. Then the series oo
Y ~ fngn
n=l converges uniformly on S.
Proof. Let f be the sum of En~176f n . Since, for n > m > 1,
n-1 Y ~ (gk+l
--
gk) = gn -- gm,
k=m
the summation by parts formula gives fkgk = Fngn
Fm-lgm
k=m
~
F t ( g k + l - gk)
f
n
gm
k=m
Y~(gk+l k=m
n-1 = (Fn -
f)gn
-
(Fm-1
-
f)gm
-
Y~ k=m
(Fk
-
f)(gk+l-
gk).
gk)
]
UNIFORM CONVERGENCE
513
By Definition A.2, given E > 0 there is an N such that IFk(s) -- f ( s ) l < E / 4 M for k > N and all s in S. Then, if n > m > N, we have k=m fkgk
< ~--~(Ignl + Igml + Ign -- gml) < ~--~2(Ignl + Igml) < E
on S. The result follows from Theorem A. 1, Q.E.D. The applicability of Abel's test is limited because we have to use another test first to determine the uniform convergence of ~-~n%l f n . This cannot be the Weierstrass M-test, (x) o~ because, if this test applies to ~ n = 1 fn, then it also applies to Y~n=l fn gn in view of the condition Ign(t)l < M. Abel's test is very useful, however, when the fn are constants rather than functions. We have the following corollary. C o r o l l a r y If Y~n=l an is a convergent series of numbers and if gn" S ---> IR are functions that satisfy conditions (ii) and (iii) of the theorem, then the series oo
~-~ an gn n=l converges uniformly on S. We are finally ready to state conditions under which continuity, integrability, and differentiability of the terms of a series of functions carry over to its sum. The first result is due to Abel who essentially proved it in 1826 but let the underlying concept of uniform convergence escape unnoticed. T h e o r e m A.4 The sum of a uniformly covergent series of continuous functions on a set S is continuous on S. P r o o f . If ~-~n= c~ 1 fn = f uniformly on S, if each fn is continuous on S, if Fn denotes the nth partial sum of the series, and if s and so are in S, then
I f ( s ) - f(s0)l < I f ( s ) - Fn(s)l + IFn(s) - Fn(sO)l + IFn(sO) - f(s0)l. By the uniform convergence of the series, the first and third terms on the fight can be made arbitrarily small by choosing n large enough. The second term can be made arbitrarily small by choosing s close enough to so. Then the left-hand side can also be made arbitrarily small by choosing s close enough to so, Q.E.D. The two remaining results are due to Weierstrass. The set S is now an interval in IR. T h e o r e m A.5 /f ~-~n%l fn is a uniformly convergent series o f integrable functions defined on an interval [a, b], then A = n=l
A n=l
that is, the sum of the series is integrable and may be integrated term by term.
514
APPENDIX A
Sn
Proof. If Fn = }--]~=l fk and if f denotes the sum of the series, let be the supremum of the set {If(x) - Fn(x)l " a < x _< b} (this concept was introduced in the proof of L e m m a A. 1). If a = x0 < xl < . . . < Xn = b is a partition of [a, b], let Mi andmi be the supremum and infimum of f on each (xi-1, xi), let Im be the supremum of the s u m s y]4n__l corresponding to f for all partitions of [a, b], and let IM n be the infimum of the sums ~ i = 1 Mi(xi - xi-1). We have
mi(xi - xi-1)
/a
(Fn - s~) <_ Im <_ l u <_
fa
(Fn + Sn),
and then 0 < IM -- Im <_ 2Sn (b - a ) . Since Sn --~ 0 as n ~ e~ because the convergence is uniform, we conclude that Im = IM. By Definition 2.2, this means that f is integrable. Finally,
f-
Fn =
( f - Fn) <_
I f - Fnl < s n ( b - a ) ,
which approaches zero as n --~ ec, Q.E.D.
Theorem A.6 /f Y]~n=l fn converges on an interval [a, b], if each fn has a continuous derivative on [a , b], and if the series ~--~n=l ~ fn' converges uniformly on [a, b], then ~-'~n~-_l fn is differentiable on [a, b] and = n=l
f~" n=l
that is, the series can be differentiated term by term. Proof. Applying Theorem A.5 to the series of derivatives, which are integrable because they are continuous, and for each x in [a, b] we obtain
faX~ n=l
=
~fa
n=l
fn' =
~-~[fn(X)
f~(a)]
n=l
=
f~(a). ~-~f n ( x ) -- n~-~ =l
n=l
Then, differentiating with respect to the upper limit of integration,
Zfn: n=l
(5)'fn n=l
on [a, b], Q.E.D. The hypotheses in Theorem A. 6 are stronger than necessary. This was done for the sake of giving a simple proof, but the conclusion is valid if the fn are assumed only to be differentiable on [a, b] and if the series Y]~n=l fn converges at a single point. 1 Note also that, in contrast with Theorems A.4 and A.5, it is the uniform convergence of Y']~n=l oo fn, that is required rather than that of Y'~n=l oo fn. The uniform convergence of Y]~n~-_l fn would not suffice, as illustrated by the following example. 1See, for instance, W. Rudin, op. cit., page 152.
EXERCISES 515
E x a m p l e A.2 The series
n:~ --~~ ~-~ sin n 3x converges uniformly on any interval by the Weierstrass M-test because the series cr
1
is known to converge. However, the series of derivatives o<3
Z n cos n3x n=l diverges at every point x - k:r, where k is an integer, because I cos n3krrl = 1, and the nth term of the series does not approach zero as n ~ cr
EXERCISES
A.1
Show that the series 1 ~n2 [sin(n3 + 7t.)(x 2 + y2) + x 3] n-1
converges uniformly on S = { (x, y) ~ IR2" x 2 + y2 < 1 }. A.2
Prove that, if a series of real numbers Y~,,~I an converges absolutely, then o@
o@ IE"''"'qL
an COSnx
and
n=l
A.3
~
converge uniformly on IR. Prove that, if a series of real numbers y-~n~=lan converges absolutely, then the series o@ n x n=l
converges uniformly on [0, ~ ) . A.4
an sin nx
n=l
)n
Show that the series o@
,,=1
converges uniformly on [e, e2].
101ogx
516
APPENDIX A
A.5
Show t h a t ~_,n~176 (xn/n !) converges uniformly on [ - a , a] for any a > 0. Is the convergence uniform on IR?
A.6
Show that the series
(x)
y~ (--1)n e-n2(x2+y2) n
n=l
converges uniformly on IR2. A.7
Show that the series (-1) n
n=~ ~
x
COS--n
converges uniformly on [ - J r / 4 , Jr/4]. oo
n
A.8
Showthat y-]~n=0(x / n ! ) converges to a continuous function f onlR. Showthat f ' = f on IR.
A.9
Show that
n=l ~ sin nx = 2 n=l (2n - 1)4 A.10
Show that for ]xl < 1 ( _ 1) n+l ~ x
log(1 + x ) = ~
n
n
n=l
(Hint:
consider the geometric series ~-]~,,~0( - x ) n ' which is uniformly convergent for Ixl
A.11
Show that the series ~
and
X 2n + 1
n=0 (2n + 1)v9
n--0
(2n)!
can be differentiated term by term and find their derivatives. On what intervals are these derivatives valid? A.12
Prove Dirichlet's test for uniform convergence: for every positive integer n, let fn : S ~ IR and gn : S --~ IR be functions such that n (i) there is a constant M such that IY]~k=l < M for each n and all s in S, (ii) gn+l (s) _< gn(S) for each n and all s in S, and (iii) gn ~ 0 uniformly on S. Then the series ~,,~--1 fngn converges uniformly on S.
A(s)l
A.13
Use Dirichlet's test to prove the uniform convergence of the series ~
1
~-~-n--1
COS
nx
n
on the interval [8, Jr - 8] for any 0 < 8 < zr/2. A.14
Prove the following corollary of Dirichlet's test for uniform convergence. For each positive integer n, let gn" S ~ lR be a function such that (i) gn+l (S) < gn(S) for each n and all s in S and (ii) gn -~ 0 uniformly on S. Then )-~n~__1( - 1)n gn converges uniformly on S.
EXERCISES
517
A.15
Define fn'[O, 1] ~ IR by fn(X) = (--x)n(1 -- X). Show that Z n ~ l fn converges uniformly on [0, 1]. Does y~n~l Ifnl converge on [0, 1]? Does it converge uniformly?
A.16
I f a > 0, define fn'[O,a] --+ IR by f ( x ) = ( - 1 ) n ( x -t-n)/n 2. Show that y~n~=l f,, oo converges uniformly on [0, a]. Does Y~n-I Ifnl converge at any x E IR? Show that the series
A.17
o~ Z(--x)n(1
--
x)e-n2y
n=l
converges uniformly on S = { (x, y) 6 IR2:0 < x < 1, y > 0 } (Hint: use Exercise A.15). A.18
Show that the series n~l =
(-1)n(xwn) k=l
converges uniformly on [0, 1] (Hint: use Exercise A. 16).