I assume you know that this is the ratio test!
Ratio test - Wikipedia, the free encyclopedia
You can use the link above or any (Advanced) Calculus book.
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.
I assume you know that this is the ratio test!
Ratio test - Wikipedia, the free encyclopedia
You can use the link above or any (Advanced) Calculus book.