Sequences and Series00(a) 'Lsinhn
1
oo (in)P
( c) L - 1 - (p an integer)
n=l n.
00(b) f)-lr1·3···(2n-1)
n=l^2 · 4· · · (2n)
oo ·n
( d) '°' L.., !'.__ nP (p real)
n=l211(e) L(-ltl/cn, where {en};'° is the Fibonacci sequence with c 0 = 0
1
omitted.- Prove:
(a) If n-+oo lim lanl/icnl = 0 and'°' L...t lcnl converges, then'°' L...t Jani converges.
(b) If n-+oo lim JanJ/ldnl = oo and'°' L...t ldnl diverges, then'°' L...t lanl diverges.
- Show that the product by Cauchy's rule of the convergent series
I:;"'(-l)n-ln-^112 by itself is a divergent series.
- Find the convergence set and sets of uniform convergence of each of
the following sequences.
(a) Un(x) = nxe-nx
2
, x E JR
(b) un(z) = zn, z E <C
(c) un(z) = nzn, z E <C(d) un(z) = nz(l - z)n, 0:::; Rez :::; 1
7. Show that the series I::= 1 (-l)n-^1 1/(n+x^2 ) converges uniformly on JR
- Show that the series I::=o x^2 /(1 + x^2 )n converges for all values of x,
yet the sum function is discontinuous at x = 0 notwithstanding the
fact that its terms are ,continuous at x = 0.
9. Show that the series I::=l x/n(l + nx^2 ) converges uniformly on JR
- Investigate the uniform: convergence of the series
00
~ (1 + nz)[l ~ (n - l)z]
- Find the region of absolute convergence of the series I::=o z^2 /(1 + z^2 )n.
- Find the region of absolute convergence and a set of uniform
convergence of
f~(z:l)n
n=l- Find the radius of convergence of the following power series.
oo n oo
(a) ~ 3n(~ +-2) (b) ~ zzn
00