Finding radius and interval of convergence for sum of x^n/ln(n+1)

[kiwifruit]

Homework Statement

i was doing this exercise and came across this example.
∞
Ʃ (x^n)/ln(n+1)
n=1

The Attempt at a Solution

i know you have to do the ratio test which is
lim | a(n+1)/a(n)|
n>∞

i got to

lim | [x ln(n+1)] /ln(n+2) |
n>∞

and have no idea how to continue? is there a way to get rid of the logs? since in this case as n>∞ the top and bottom just goes to ∞?

 

[susskind_leon]

ln(n)/ln(n+1) goes to 1 for n -> infty
You can prove that easily by proving that ln(n)/ln(n+1)-1 approaches zero.

 

[HallsofIvy]

Or use L'Hopital's rule: the derivative of ln(x+1) is 1/(x+1) and the derivative of ln(x) is 1/x so this becomes (1/x)(x+1/1)= (x+1)/x goes to 1 as x goes to infinity.

That shows that ln(x)/ln(x+1) goes to 1 as x goes to infinity which includes n going to infinity.