212
oo I
(e) "'"'2 ;_,J !::_Zn nn
n=l(^00) ( I )2
(g) L (~~ )! Zn
n=l
(
i) ~ 22.42···(2n)2 Zn
~ 32 · 52 · · · (2n + 1)^2
000(^00) ( I )3
(h) "'\' ~ n
~ (3n)! z
( j ) 2 -z+···+^1 2n.3n-l^1 z 2n-1 + --z 2n.3n^1 Zn +···
(k) f (-~r zn(n+l)
n=l
Chapter4
In part (k) determine whether or not the series converges for z = i.
If the series converges, find its sum.
- Find the region of absolute convergence of each of the following.
oo 1 oo 3n
(a) L2n+1+1(z-1r (b) L5n+1(z+2r
n=O n=O
00 00
.,.---. 3 "'\'
(c) 2.,qn (z-3t (lql < 1) (d) L..,nP(z-2t
0 1
(e) f 1 ;:2n Cf) f (2:zf
n=O n=O- Suppose that the radius of convergence of :Z:::::'=o anzn is R. Find the
radius of convergence of each of the following series.
00 00
(a) :Sanz^2 n (b) L:a;zn
oo^0 oo·^0
(c) L anzn (d) L nkanzn
0 0- Find the sum function of the series :Z:::;;"'[cos(2mr/3)]zn
17. Show that if :Z:::;;"' an·Zn converges for z = z 0 , it converges absolutely for
every z such that lzl < Jz 0 J. If the series diverges for z = z 0 , it diverges
for every z such that Jzl > lzol (Abel's theorem).
18. If I;;;'° anzn has radius of convergence R -j. 0, show that the series
converges uniformly on every compact set ]{ contained in lzl < R.- Prove that if :Z:::;;"' anzn has radius of convergence R (0 < R < oo)
and the series converges absolutely for z = ReiBo, then it converges
absolutely and uniformly on JzJ :S .R.
20. If :Z:::;;"' anzn has radius of convergence R1 and :Z:::;;"' bnzn has radius of
convergence R 2 , prove the following.
00
(a) The radius of convergence of L(an + bn)zn is R = min(R1,R2).
0