Question

For what values of x does this series converge?

Answer 1

geometric series converge when |r| < 1.
So |ln(x)| < 1. solve for x

Answer 2

But still, you can consider the convergence conditions for the series \(\sum x^n\).

Answer 3

https://www.wolframalpha.com/input/?i=solve+abs%28ln%28x%29%29+%3C+1

Answer 4

you still have to check the endpoints.
so it's actually 1/e <= x < e

Answer 5

use ratio test
S=(lnx)^n/(lnx)^n+1=1/lnx
for convergency that fraction must be less than 1.

Answer 6

thank you! :D