Question

Find the radius of convergence of the series:

Answer 1

\[\sum_{n=1}^{\infty} \frac{ x^n }{ n9^n }\]

Answer 2

Already did ratio test: \[\lim_{n \rightarrow \infty} =\frac{ |x| }{ 9 }\], therefore the series converges when |x|/9=0

I already have my endpoints done and checked, but I'm having trouble finding R

Answer 3

x^(n+1) (n) 9^(n)
-------------------
x^(n) (n+1) 9^(n+1)


x (n)
------
9 (n+1)


|x/9| lim n/(n+1) = |x/9|

the radius of convergence is half the interval of convergence

|x/9| < 1

-1 < x/9 < 1

-9 < x < 9

since the interval is 18, whats the radius?

Answer 4

and to check the endpoints, just let x=-9, or 9 and see if it limits