Answer to Question #314403 in Real Analysis for Pankaj

Question

Test the following series for convergence.

Σ from n=1 to ♾️ for n.x^n-1 , x>0

Answer 1

$R=\frac{1}{\underset{n\rightarrow \infty}{\lim}\sqrt[n]{n}}=\frac{1}{1}=1;\\x=-1:\sum_{n=1}^{\infty}{n\left( -1 \right) ^n}diverges\,\,because\,\,\underset{n\rightarrow \infty}{\lim}n\left( -1 \right) ^n=\infty; \\x=1:\sum_{n=1}^{\infty}{n\cdot 1^n}diverges\,\,because\,\,\underset{n\rightarrow \infty}{\lim}n=\infty; \\x\in \left( -1,1 \right) -interval\,\,of\,\,convergence$