Let sum(a_nx^n) be a power series with a_n not zero, and assume L=lim|a_(n+1)/a_n| exists.

a) Show that if L is not zero, then the series converges for all x in (-1/L,1/L).

b) Show that if L=0, then the series onverges for all x in R

c) Show that a) and b) continue to hold if L is replaced by the limit

L'=lim(s_n) where s_n=sup{|a_(k+1)/a_k|:k>=n}

The value L' is called the limit superior or lim sup of the sequence |a_(n+1)/a_n|. It exists iff the sequence is bounded.

d) Show that if |a_(n+1)/a_n| is unbounded, then the original series converges only when x=0

I'm looking at this and I have no clue how to start.

Like for a), I start by assuming L is not zero.

So we have lim|a_(n+1)/a_n|is nonzero. Then I have trouble getting further.