I need to find the interval on convergence of the given power series
I am trying to do summation of nx^x
I am using s(n+1)/s(n) to solve the problem, but am having trouble getting an answer. Thanks for the help.
We wish to find the radius of convergence for $\displaystyle \sum_{n = 1}^{ \infty} nx^n$
We will proceed by the Ratio test (since that's what you wanted, I'd probably do the root test here).
Let $\displaystyle s_n = nx^n$
by the Ratio test, we have convergence if $\displaystyle \lim_{n \to \infty} \left| \frac {s_{n + 1}}{s_n}\right| < 1$
Now, $\displaystyle \lim_{n \to \infty} \left| \frac {(n + 1)x^{n + 1}}{nx^n} \right| = \lim_{n \to \infty} \left| \frac {n + 1}{n} \cdot x \right| = |x|$
Thus we have convergence if $\displaystyle |x| < 1$
so our radius of convergence is 1