Does Convergence Rate Affect Series Behavior?

[jkh4]

Let (infinity)(sigma)(n=1) = An be a series with positive terms such that lim(n -> infinity) = (An+1)/(An) = L < 1

a) Let L < r < 1. Show that there is an N > 0 such that for all n > N, we have (An+1)/(An) < r

b) Show that Ak+N < or = ANr^k for k = 1, 2...

c) Show that lim (k -> infinity) (Ak+N)^(N+k) < or = r

Thanks!

For An+1, it's A with sub n+1
An, it's A sub n
Ak+N is A such (k+N)
ANr^k is A sub N times r^k

Thanks!

 

[HallsofIvy]

Let (infinity)(sigma)(n=1) = An be a series with positive terms such that lim(n -> infinity) = (An+1)/(An) = L < 1

a) Let L < r < 1. Show that there is an N > 0 such that for all n > N, we have (An+1)/(An) < r

Use the DEFINITION of "limit of a sequence".

b) Show that Ak+N < or = ANr^k for k = 1, 2...

Proof by induction on k.

c) Show that lim (k -> infinity) (Ak+N)^(N+k) < or = r

After b, this should be obvious. What is the limit of r^N+k?