Thread: Convergence of a positive series - true or false question

1. Convergence of a positive series - true or false question

True/False:
If a positive series $\Sigma a_n$ converges then it must be that $\lim_{n\to \infty} \sqrt[n]{a_n} \leq 1$.

According to the textbook the answer is false, but I can't find a counterexample. Any ideas?

2. Re: Convergence of a positive series - true or false question

Choose

$a_n=\begin{Bmatrix} 0 & \mbox{ if }& n\mbox{ even}\\\dfrac{1}{n^2} & \mbox{if}& n\mbox{ odd}\end{matrix}$

then, $\sum_{n=0}^{+\infty}a_n$ is convergent (compare with $\sum_0^{+\infty}1/n^2$), however does not exist $\lim_{n\to +\infty}\sqrt[n]{a_n}$.