1. Consider the power series P∞
k=0 ak(x−x0)k.
(a) Show that radius of convergence of the series is ∞if lim
n→∞
an+1
an
= 0.
(b) Show that radius of convergence of the series is 0 if lim
n→∞
an+1
an
=∞.
2. Consider the power series P∞
k=0 ak(x−x0)k.
(a) Show that radius of convergence of the series is ∞if lim sup
n→∞
n
p|an|= 0.
(b) Show that radius of convergence of the series is 0 if limsup
n→∞
n
p|an|=∞.
3. Suppose {an}is defined by:
an=n−nif nis odd
2−nif nis even
Find the radius of convergence of P∞
k=0 ak(x−3)k
1