- 7 Analytic Functions 529
k (k = 0, 1, 2, 3,-· · ) is
[j~O e;j)ak+j(d-c)jl (x-d)k.
(We know that t he column sums exist because they are subseries of an abso-
lutely convergent series.)
By Theorem 8.7.16, the sum of these column sums must be f(x). Therefore,
f(x) = ~^00 [ _f;^00 (k + j J ") ak+J(d - c)^1 l (x - dl.
Since this is true whenever Ix -di < p-lc - di, the radius of convergence is at
least p - I c - di- •
Theorem 8.7. 13 is of little practical computational value. For example, try
applying it to the Maclaurin series for ln(l + x) to calculate the Taylor coeffi-
cients bk in the Taylor series for ln(l + x) about c = 1 /2.
*DOUBLE SERIES
00
Definition 8.7.14 The notation I: aij is used to represent the "sum" of all
i,j=l
the entries in the infinite matrix
au a12 a13 a1j
a21 a22 a2 3 · · · a2j
a31 a32 a33 · · · a3j
(35)We shall not concern ourselves with all the various ways one can define such
a sum, but shall focus on only two. We define the row sums R 1 , R2, · · · , ~, · · ·
and column sums C 1 , C2, · · · , C1, · · · of (35) by
00 00
Ri = I: aij and Cj = I: aij·
j=l i=l
If all the row sums (series) R i converge, then the series