1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 557

so that

n+p-1

+ L (ak - ak+i)lzlk+l + an+plzjn+p+l

k=n+l

For jzj = 1 but z f= 1, we get

IRn,p(z)j:::; jz ~ lj [~n+~ + nf \ak - ak+1) + an+p]

k=n+l

which holds for all p ~ 1. Hence we can let p --+ oo, to obtain

But an+i --+ 0 as n --+ oo, and it follows that Rn(z) --+ 0 as n --+ oo
whenever jzj = 1, z f= 1. So the power series converges under the stated
conditions. However, for z = 1 the series may converges or diverge as
examples I::'= 1 (1/n^2 )zn, l.:::'= 1 (1/n)zn show.

Corollary 8.6 Under the assumptions of Theorem 8.14, the series

l:(-l)nan converges.

Proof It suffices to set z = -1 in I: anzn. This particular case of Picard's
theorem is called the Leibniz test for alternating series of real terms.

Theorem 8.15 (Abel's Theorem). If I: anzn has a radius of convergence
R = 1 and the series converges for z = 1, so that I: an = S, then

00

f(z) = L anZn __, S

n=O

as z --+ 1 along any path lying between two equal chords of the unit circle

passing through z = 1.

Remarks Such an angle is sometimes called a Stolz angle corresponding

to the point z = 1.

Theorem 8.15 is a generalization of Abel's original theorem which was

established for real power series (the limit in this case taken as x--+ 1-).