I should calculate the radius of convergenc and would like to know, if the result $\frac{1}{3}$ is correct.
Here the exercise:
$ \frac{x}{1\cdot3} + \frac{x^2}{2\cdot3^2} + \frac{x^3}{3\cdot3^3} + \frac{x^4}{4\cdot3^4}... $
This is: $ \sum\limits_{n=0}^\infty \frac{x^{n+1}}{(n+1)3^{n+1}} \\ \lim\limits_{n \to \infty} \left| \frac{(n+1)\cdot3^{n+1}}{(n+2)\cdot3^{n+2}} \right| = \left| \frac{1}{3} \right| $
Iām right? Thanks.
Summery
I could test with the ratio test if a power series is convergent. I could use $\lim\limits_{n \to \infty} \frac{|a_{n+1}|}{\left|a_{n}\right|}$ and get the $\left|x\right|$ for which the series is convergent. With that test the series is convergent, if the result is $<1$.