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The summation of power series having positive coefficients
Applied Numerical Mathematics 4 (1988) 471-476 North-Holland
471
THE SUMMATION OF POWER SERIES HAVING POSITIVE COEFFICIENTS A.M. COHEN Department of Mathematics, UWIST, Cardiff CFI 3EU. United Kingdom
I.M. LONGMAN Department of Geophysics & Planetary Sciences, Faculty of Exact Sciences, Tel-Aviv University, Ramat-Avrv 69978, Tel-Aviv, Israel
The paper develops a method for the summation of a series of positive terms given by Gismalla et al. [2]. Here we are concerned with the summation of power series CT=Ip,xn where CL,,>0 and O
1. Alternative formulation of the method of Gismalla, Jenkins and Cohen We assume that the pk are moments
* Pu,= / 0u
k-lf(u)du,
k=l,2,3,
...
(1)
of a function f(u) defined in (0,l) which is such that the integrals (1) exist as Stieltjes integrals, and also such that the Stieltjes integral I
=
xb)
’
J -l-ux
(2)
du
0
exists. Of course the p, as defined in (1) may not be all of one sign, although it is most unlikely that they will be of alternating signs. However the importance of this paper is for monotonic series, which are very difficult to accelerate, and it will be assumed henceforth in this paper that the pk are positive. If 0 < x < 1, then the integrand in (2) can be expanded so that I = X/f(u)
du + x211uf (u) du + x3/‘u2f
0
0
0
=prX+jL2X2+/h3X3+
(u) du +
9 a 9
‘**,
i.e. I represents a power series whose coefficients are the moments produce a reformulation of (3) starting from the identity -=-1 k-U
0168-9274/88/$3.50
1 2u 1+u+1-u2* 0 1988, Elsevier Science Publishers B.V. (North-Holland)
(3)
pl, p2.. . . We shall now
(4)
A.M. Cohen, I.M. Lmgman / Summation of power series
472
Applying the identity to l/(1 - u2) gives -=
1
1 _+2ul+u
1-U
and application
2u2 1 -=_ + lBU4]
1 [ 1+u2
1 l+u
+ - 2u 1+u2
+ _ 4u3 l-u4
(5)
to l/(1 - u4) yields
-=-1 1-U
1 1+u+
2u 1+u2+
- 4u3 Mu4
_* (6)
+ - gu7 l-zig’
and so on. In particular, replacing u by ux, we have the more general result -- 2ux 1 + M + 1 + &2 1
-=- 1 1-u.x
+ 2n-yu)2”-‘-1
4u3x3 +
+
1 +u4x4
.*=
1 + ( tLx)2”-’
+ 2”(42”-’ 1 - (l.&
*
(7) Substituting
for l/(1 - u.x) in (2) we have
1
lfb) du +
=x
I()
zx2
l+ux
’ uf (u) du + . . . I 0 1 +u2x2
+y-y-’
1 u2”-‘-‘f I
(u) du
1 + ( ux)2”-’
O
1 u2=‘f (u) du + 2”xZn I 0 1-(ux)‘” ’
09
i.e. I=&+
(9)
Wm.+S,+R,
where
(10) (11)
S,=~lX-~2X2+p3X3-'*.,
2(&X2
s, =
p4x4 + jx6XZ-
l
..
1
and, in general, Sn=2n-1(P2n_~X2"-' -P2.2n_~X2'2"-'+/&3_2ti-lX3'2"-'-m ) l
l
(12)
and R =
2”x2n
1 u2=‘f(u) /0
l-(U)”
du -
(13)
We shall be concerned with power series in x where 0 < x < 1 and x may be very close to (but not equal to) 1. It will be seen that with x = 1 the alternating series S,, S,, . . . are identical with those given by Gismalla et al. f2]. R can be thought of as a truncation error term and in the next section we will demonstrate how the truncation error can be reduced by the additional computation of a small number of terms. The computation of the Si to a high degree of accuracy can be achieved in a variety of ways e.g. by the method of Longman [4], Levin [3], the iterated Shanks method (Shanks [5], Cohen [l]), etc. It may be remarked that as i increases direct summation of a finite number of terms will be sufficient to determine Si accurately.
A.M. Cohen, I. M. Longman / Summation of power series
473
2. “Low-cost” improvement of accuracy Consider the term R in (13). It can be written as R = 2”
1 u’“--‘f (u) du J0
A -u2”
(14)
’
where A = xm2” 1s ’ greater than 1 if 0 < x < 1. Write V = u2” and consider the term l/( A - V). It is well known that we can express l/( A - V) in the form 1 -=ia,+a,T,*(V)+ A-V
L
+akTk*(V)+
where Tk*(V) is the shifted Chebyshev polynomial
%Tk*+*( A vv) ,
of the first kind of degree k in [0,11,and
rk = 1/T,*,,(A)*
(16)
For any value of k we can determine by (A - V), then using the identity
a,, aI,. . . , ok by multiplying
both sides of equation (15)
and finally by comparing the coefficients of T,*( V), 0 < r G k + 1, on both sides of the resuiting equation. For ail k (>, i) we have ak-i=
(4A - 2)ak_i+l-
ak_i+2,
i >, 2
(18)
and when k = 3, for example, we have a, = (256A3 - 384A2 + 160A - l6)~3, a1 = (64~ 2 - 64A + 12)~,,
a2 = (16~ - 8)r3,
1
a3 = 4r3,
1
” = (128A4 - 256A3 + 160A2 - 32A + 1) = T4*(A) ’
09)
Hence from (14), (15) and (19) we have R=2”
J(ia0 + a,(2u2” *
1) + a2(8u2’2” - 8~‘” + 1)
0
+ a3(32u3’2” - 48~~~~”=t 18~~” - 1))
u’“-‘f (u) du + R;,
(20)
where Rj4’
1 r u2”-‘T,*(u’“)f(u) 3
J
A
0
--2”
du -
Thus R = 2”[($ao - a, + a2 - a3)p2n + (2a, - 8a2 + 18a3)p2.~ll + @a, - 48a3)p3.2n + 32a3p4.,.] = T3+ R;,
say
+ R;
(21)
A.M. Cohen, I.M. Longman / Summation of power series
474
Table 1 Coefficients in the expansion (15) where k = 4, 5, 6 i
ak-i
0
47k
1
2 3 4 5 6
(16A - 8)7k (#A2 -64A 12)Tk (256A3- 384A2 + 160A - 16)Q 9 13#A2 - 320A + 2O)Tk (10L4A4 -20aA3 (4096A’ - 10240A4 + 9216A3 - 3584A2 + 560A - 24)rk (16384A6 -49152A’ + 56320A4- 30720A3+ 8064A2-896A + 28)?, r4= l/(512 As - 1280A4+ 1120A3-400A2 +50A - 1) = l/T,*( A) =1/T,*(A) rs = 1/(2048A6-6144As +6912A4 -3584A3 +84OA’-72A+l) q, = l/(8192 A’ - 672A6 + 39424A’ - 26880A4 + 9408A3 - 1568A2 + 98A - 1) = l/T,* (A)
and we can
definitely assert that
I&I
(22)
<73lRl
as 17’,‘,*(u2”) I < 1, but
< 1
at most points in (0, l] so that the strict inequality (22) holds. The importance of this result is that just by forming T3, a linear combination of the coefficients, p2n, of the given series (3) we can improve on the accuracy. For instance, with c12.25 p3-2”, tr4.2n, A = 10, it follows from (19) that 73< 10e6 and thus by the addition of T3 to Cy=rSi the truncation error is cut to less than one millionth of its former value. Similarly when k = 4, 5, 6 we obtain the ai from Table 1 (entries only appear if k - i >, 0). Table 1 can readily be extended by the use of (16) and (18). It follows that R=T,+R;,
k=4,5,6
where T4= ?[($a,
- a, + a2 - a3 + a,)pzn + (2a, - 8a2 + 18a3 - 36a,)p2.2n
+ (8a, - 48a3 + 160a,)p3.2n + (32a3 - 256a,)p,.2n
+ 128a,ps.2n],
T, = 2” [ ( ia0 - aI + a2 - a3 + a4 - a,)pzn -I-(2a, - 8a2 + 18a3 - 32a, + 50a,)p2.2n + ($a, - 48a, + 160a, - 4OOa,)p3.y + (32a, - 256a, + P120as)p4.2n + (128~~~- 1280a5)p5.2n + 512a5p62n] and & = 2" [ ( ia0 - a1 + a2 - a3 + a4 - a5 + a,)p2n
+ (2a, - 8a, + 18a, - 32a, + 5Oa, - 72a,)p2.2n i- (8a, - 48a, + i60a, - 400a, + 840a,)p3.2n -t (32a, - 256a, + 1120a, - 3584a&,.2n -t-(128a, - 1280a, + 6912as)p5.2n + (512a, - 6144a,)p,.2n
+ 2048ahp7.2n],
A.M. Cohen, I. M. Longman / Summation of power series
475
where also R; = 2”~
1U2”-1Tk*+l(~2”)f(U) du, /0 A - u2”
k= 4 9 5 96 .
3. Numerical results and conclusions We now consider the application of the previous two sections to concrete examples. For this purpose we have chosen two examples for which Levin’s u-transform was unsuccessful (see [l]). Example 3.1. ET’ ,(0.9999)“/n (Exact answer := In 10000 = 9.2103403719 76.. . ). In Table 2 we have computed the series S,, S2, . . . until a stage is reached when in the general case, x2” (= l/A) is less than some given value B. We then compute, for chosen k, the correction term Tk and the error estimator rk. The bold sum S15 has only been used for k = 3. We have used 229 Table 2 Estimation of Zz_,(0.9999)n/n
n
Sn (rounded) 1 2 3 4 5 6
7 8 9 10 11 12 13 14 15
0.69309 0.69304 0.69294 0.69274 0.69234 0.69154 0.68995 0.68676 0.68042 0.66787 0.64325 0.59597 0.50916 0.36517 0.13753
71793 71805 71905 72405 74605 83806 21410 73424 84664 35774 48990 63256 55459 49759 90907
09904 60112 61279 66610 89247 82077 52585 46962 72715 53312 96549 71828 42989 10866 50157
B A CSi rk
Correction sum
k=3
k=6
0.1 26.49520 70586 71339 $17186 69970 67191 1.712x lo-* 0.03847 33749 52915 9.21034 03720 20
0.2 5.14734 95178 8.99432 79063 2.661 x 1O-9 0.21601 24656 9.21034 03719
Number of correct figures is underlined.
Table 3 Values of A required to make ?k < lo-’
(r = 6,7,8)
rk
lO+j
IO-’
1o-8
9.91 5.07 3.33 2.52
17.23 7.73 4.64 3.29
30.24 11.94 6.56 4.36
26756 17034 60537 77571
A.M. Cohen, t. M. Longman / Summation of power series
476
Table 4 Estimation of CT_,(0.99)n ln(n + l)/(n n
+ l)* k=3
S, (rounded 1 0.10072 0.16093 0.18092 0.15432 0.10699 0.06204 0.02907 0.00959
84855 27108 57264 34458 98545 71673 93512 92771
69082 49966 26619 77209 16207 42332 13306 42004
B A =I
0.1 13.10358 65934 7886 0.80463 60189 36727 3.101 x lo-’ 0.00167 80535 40021 0.80631 40724 767
7k
Correction Sum
terms of the series to obtain the sum with k = 3 and 217 terms with k = 6. Table 3 provides details of the size of A required to make T/,< 10B6, lo-’ and lo-* for k = 3,. . . ,6. Example 3.2. XT_ ,(0.99)“ln( n + l)/( n + l)* ( = 0.8063140724 29.. . ). Proceeding as with Example 3.1. we obtain Table 4. 124 terms of the series have been used in the computation of the above sum. The numerical results confirm the value of the method since, in Table 2 for example, the addition of just 4 mole computed terms to Zir,si increases the accuracy from 1 correct significant digit to 9 (in fact the answer is correctly rounded to 10 decimal places) while adding ‘7 more terms to Xi$.,si increases the accuracy from no correct significant figures to 12.
Acknowledgment AMC whishes to thank the Royal Society and UWIST for making it possible for him to visit Tel-Aviv University. The authors wish to thank the referee for helpful comments which have served to improve the presentation in this paper.
References Cohen, On improving convergence of alternating series. Intcrnat. J. Comput. Math. B 9 (1981) 45-53. PI A.M. ,
121 D.A. Cismalh, L.D. Jenkins and A.M. Cohen, Acceleration of convergence of series with application to the evaluation of certain multiple integrals, Internat. J. Comput. Math. (to appear). VI D. Levin, Development of non-linear transformations for improving convergence of sequences, Internat. J. Comput. Math. B 3 (1973) 371-388. 141 I.M. Longman, The summation of series, Appl. Numer. Math. 2 (1986) 135-141. PI D. Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) l-42.
471
THE SUMMATION OF POWER SERIES HAVING POSITIVE COEFFICIENTS A.M. COHEN Department of Mathematics, UWIST, Cardiff CFI 3EU. United Kingdom
I.M. LONGMAN Department of Geophysics & Planetary Sciences, Faculty of Exact Sciences, Tel-Aviv University, Ramat-Avrv 69978, Tel-Aviv, Israel
The paper develops a method for the summation of a series of positive terms given by Gismalla et al. [2]. Here we are concerned with the summation of power series CT=Ip,xn where CL,,>0 and O
1. Alternative formulation of the method of Gismalla, Jenkins and Cohen We assume that the pk are moments
* Pu,= / 0u
k-lf(u)du,
k=l,2,3,
...
(1)
of a function f(u) defined in (0,l) which is such that the integrals (1) exist as Stieltjes integrals, and also such that the Stieltjes integral I
=
xb)
’
J -l-ux
(2)
du
0
exists. Of course the p, as defined in (1) may not be all of one sign, although it is most unlikely that they will be of alternating signs. However the importance of this paper is for monotonic series, which are very difficult to accelerate, and it will be assumed henceforth in this paper that the pk are positive. If 0 < x < 1, then the integrand in (2) can be expanded so that I = X/f(u)
du + x211uf (u) du + x3/‘u2f
0
0
0
=prX+jL2X2+/h3X3+
(u) du +
9 a 9
‘**,
i.e. I represents a power series whose coefficients are the moments produce a reformulation of (3) starting from the identity -=-1 k-U
0168-9274/88/$3.50
1 2u 1+u+1-u2* 0 1988, Elsevier Science Publishers B.V. (North-Holland)
(3)
pl, p2.. . . We shall now
(4)
A.M. Cohen, I.M. Lmgman / Summation of power series
472
Applying the identity to l/(1 - u2) gives -=
1
1 _+2ul+u
1-U
and application
2u2 1 -=_ + lBU4]
1 [ 1+u2
1 l+u
+ - 2u 1+u2
+ _ 4u3 l-u4
(5)
to l/(1 - u4) yields
-=-1 1-U
1 1+u+
2u 1+u2+
- 4u3 Mu4
_* (6)
+ - gu7 l-zig’
and so on. In particular, replacing u by ux, we have the more general result -- 2ux 1 + M + 1 + &2 1
-=- 1 1-u.x
+ 2n-yu)2”-‘-1
4u3x3 +
+
1 +u4x4
.*=
1 + ( tLx)2”-’
+ 2”(42”-’ 1 - (l.&
*
(7) Substituting
for l/(1 - u.x) in (2) we have
1
lfb) du +
=x
I()
zx2
l+ux
’ uf (u) du + . . . I 0 1 +u2x2
+y-y-’
1 u2”-‘-‘f I
(u) du
1 + ( ux)2”-’
O
1 u2=‘f (u) du + 2”xZn I 0 1-(ux)‘” ’
09
i.e. I=&+
(9)
Wm.+S,+R,
where
(10) (11)
S,=~lX-~2X2+p3X3-'*.,
2(&X2
s, =
p4x4 + jx6XZ-
l
..
1
and, in general, Sn=2n-1(P2n_~X2"-' -P2.2n_~X2'2"-'+/&3_2ti-lX3'2"-'-m ) l
l
(12)
and R =
2”x2n
1 u2=‘f(u) /0
l-(U)”
du -
(13)
We shall be concerned with power series in x where 0 < x < 1 and x may be very close to (but not equal to) 1. It will be seen that with x = 1 the alternating series S,, S,, . . . are identical with those given by Gismalla et al. f2]. R can be thought of as a truncation error term and in the next section we will demonstrate how the truncation error can be reduced by the additional computation of a small number of terms. The computation of the Si to a high degree of accuracy can be achieved in a variety of ways e.g. by the method of Longman [4], Levin [3], the iterated Shanks method (Shanks [5], Cohen [l]), etc. It may be remarked that as i increases direct summation of a finite number of terms will be sufficient to determine Si accurately.
A.M. Cohen, I. M. Longman / Summation of power series
473
2. “Low-cost” improvement of accuracy Consider the term R in (13). It can be written as R = 2”
1 u’“--‘f (u) du J0
A -u2”
(14)
’
where A = xm2” 1s ’ greater than 1 if 0 < x < 1. Write V = u2” and consider the term l/( A - V). It is well known that we can express l/( A - V) in the form 1 -=ia,+a,T,*(V)+ A-V
L
+akTk*(V)+
where Tk*(V) is the shifted Chebyshev polynomial
%Tk*+*( A vv) ,
of the first kind of degree k in [0,11,and
rk = 1/T,*,,(A)*
(16)
For any value of k we can determine by (A - V), then using the identity
a,, aI,. . . , ok by multiplying
both sides of equation (15)
and finally by comparing the coefficients of T,*( V), 0 < r G k + 1, on both sides of the resuiting equation. For ail k (>, i) we have ak-i=
(4A - 2)ak_i+l-
ak_i+2,
i >, 2
(18)
and when k = 3, for example, we have a, = (256A3 - 384A2 + 160A - l6)~3, a1 = (64~ 2 - 64A + 12)~,,
a2 = (16~ - 8)r3,
1
a3 = 4r3,
1
” = (128A4 - 256A3 + 160A2 - 32A + 1) = T4*(A) ’
09)
Hence from (14), (15) and (19) we have R=2”
J(ia0 + a,(2u2” *
1) + a2(8u2’2” - 8~‘” + 1)
0
+ a3(32u3’2” - 48~~~~”=t 18~~” - 1))
u’“-‘f (u) du + R;,
(20)
where Rj4’
1 r u2”-‘T,*(u’“)f(u) 3
J
A
0
--2”
du -
Thus R = 2”[($ao - a, + a2 - a3)p2n + (2a, - 8a2 + 18a3)p2.~ll + @a, - 48a3)p3.2n + 32a3p4.,.] = T3+ R;,
say
+ R;
(21)
A.M. Cohen, I.M. Longman / Summation of power series
474
Table 1 Coefficients in the expansion (15) where k = 4, 5, 6 i
ak-i
0
47k
1
2 3 4 5 6
(16A - 8)7k (#A2 -64A 12)Tk (256A3- 384A2 + 160A - 16)Q 9 13#A2 - 320A + 2O)Tk (10L4A4 -20aA3 (4096A’ - 10240A4 + 9216A3 - 3584A2 + 560A - 24)rk (16384A6 -49152A’ + 56320A4- 30720A3+ 8064A2-896A + 28)?, r4= l/(512 As - 1280A4+ 1120A3-400A2 +50A - 1) = l/T,*( A) =1/T,*(A) rs = 1/(2048A6-6144As +6912A4 -3584A3 +84OA’-72A+l) q, = l/(8192 A’ - 672A6 + 39424A’ - 26880A4 + 9408A3 - 1568A2 + 98A - 1) = l/T,* (A)
and we can
definitely assert that
I&I
(22)
<73lRl
as 17’,‘,*(u2”) I < 1, but
< 1
at most points in (0, l] so that the strict inequality (22) holds. The importance of this result is that just by forming T3, a linear combination of the coefficients, p2n, of the given series (3) we can improve on the accuracy. For instance, with c12.25 p3-2”, tr4.2n, A = 10, it follows from (19) that 73< 10e6 and thus by the addition of T3 to Cy=rSi the truncation error is cut to less than one millionth of its former value. Similarly when k = 4, 5, 6 we obtain the ai from Table 1 (entries only appear if k - i >, 0). Table 1 can readily be extended by the use of (16) and (18). It follows that R=T,+R;,
k=4,5,6
where T4= ?[($a,
- a, + a2 - a3 + a,)pzn + (2a, - 8a2 + 18a3 - 36a,)p2.2n
+ (8a, - 48a3 + 160a,)p3.2n + (32a3 - 256a,)p,.2n
+ 128a,ps.2n],
T, = 2” [ ( ia0 - aI + a2 - a3 + a4 - a,)pzn -I-(2a, - 8a2 + 18a3 - 32a, + 50a,)p2.2n + ($a, - 48a, + 160a, - 4OOa,)p3.y + (32a, - 256a, + P120as)p4.2n + (128~~~- 1280a5)p5.2n + 512a5p62n] and & = 2" [ ( ia0 - a1 + a2 - a3 + a4 - a5 + a,)p2n
+ (2a, - 8a, + 18a, - 32a, + 5Oa, - 72a,)p2.2n i- (8a, - 48a, + i60a, - 400a, + 840a,)p3.2n -t (32a, - 256a, + 1120a, - 3584a&,.2n -t-(128a, - 1280a, + 6912as)p5.2n + (512a, - 6144a,)p,.2n
+ 2048ahp7.2n],
A.M. Cohen, I. M. Longman / Summation of power series
475
where also R; = 2”~
1U2”-1Tk*+l(~2”)f(U) du, /0 A - u2”
k= 4 9 5 96 .
3. Numerical results and conclusions We now consider the application of the previous two sections to concrete examples. For this purpose we have chosen two examples for which Levin’s u-transform was unsuccessful (see [l]). Example 3.1. ET’ ,(0.9999)“/n (Exact answer := In 10000 = 9.2103403719 76.. . ). In Table 2 we have computed the series S,, S2, . . . until a stage is reached when in the general case, x2” (= l/A) is less than some given value B. We then compute, for chosen k, the correction term Tk and the error estimator rk. The bold sum S15 has only been used for k = 3. We have used 229 Table 2 Estimation of Zz_,(0.9999)n/n
n
Sn (rounded) 1 2 3 4 5 6
7 8 9 10 11 12 13 14 15
0.69309 0.69304 0.69294 0.69274 0.69234 0.69154 0.68995 0.68676 0.68042 0.66787 0.64325 0.59597 0.50916 0.36517 0.13753
71793 71805 71905 72405 74605 83806 21410 73424 84664 35774 48990 63256 55459 49759 90907
09904 60112 61279 66610 89247 82077 52585 46962 72715 53312 96549 71828 42989 10866 50157
B A CSi rk
Correction sum
k=3
k=6
0.1 26.49520 70586 71339 $17186 69970 67191 1.712x lo-* 0.03847 33749 52915 9.21034 03720 20
0.2 5.14734 95178 8.99432 79063 2.661 x 1O-9 0.21601 24656 9.21034 03719
Number of correct figures is underlined.
Table 3 Values of A required to make ?k < lo-’
(r = 6,7,8)
rk
lO+j
IO-’
1o-8
9.91 5.07 3.33 2.52
17.23 7.73 4.64 3.29
30.24 11.94 6.56 4.36
26756 17034 60537 77571
A.M. Cohen, t. M. Longman / Summation of power series
476
Table 4 Estimation of CT_,(0.99)n ln(n + l)/(n n
+ l)* k=3
S, (rounded 1 0.10072 0.16093 0.18092 0.15432 0.10699 0.06204 0.02907 0.00959
84855 27108 57264 34458 98545 71673 93512 92771
69082 49966 26619 77209 16207 42332 13306 42004
B A =I
0.1 13.10358 65934 7886 0.80463 60189 36727 3.101 x lo-’ 0.00167 80535 40021 0.80631 40724 767
7k
Correction Sum
terms of the series to obtain the sum with k = 3 and 217 terms with k = 6. Table 3 provides details of the size of A required to make T/,< 10B6, lo-’ and lo-* for k = 3,. . . ,6. Example 3.2. XT_ ,(0.99)“ln( n + l)/( n + l)* ( = 0.8063140724 29.. . ). Proceeding as with Example 3.1. we obtain Table 4. 124 terms of the series have been used in the computation of the above sum. The numerical results confirm the value of the method since, in Table 2 for example, the addition of just 4 mole computed terms to Zir,si increases the accuracy from 1 correct significant digit to 9 (in fact the answer is correctly rounded to 10 decimal places) while adding ‘7 more terms to Xi$.,si increases the accuracy from no correct significant figures to 12.
Acknowledgment AMC whishes to thank the Royal Society and UWIST for making it possible for him to visit Tel-Aviv University. The authors wish to thank the referee for helpful comments which have served to improve the presentation in this paper.
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121 D.A. Cismalh, L.D. Jenkins and A.M. Cohen, Acceleration of convergence of series with application to the evaluation of certain multiple integrals, Internat. J. Comput. Math. (to appear). VI D. Levin, Development of non-linear transformations for improving convergence of sequences, Internat. J. Comput. Math. B 3 (1973) 371-388. 141 I.M. Longman, The summation of series, Appl. Numer. Math. 2 (1986) 135-141. PI D. Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) l-42.