# Give an example of a convergent series

#### Alexmahone

##### Active member
Give an example of a convergent series $\sum a_n$ for which $(a_n)^{1/n}\to 1$.

PS: I think I got it: $\sum\frac{1}{n^2}$

#### CaptainBlack

##### Well-known member
Give an example of a convergent series $\sum a_n$ for which $(a_n)^{1/n}\to 1$.

PS: I think I got it: $\sum\frac{1}{n^2}$
This checks out numerically.

Note we are already restricted to series the terms of which are eventually all positive.

Now: $$(a_n)^{1/n}\to 1$$ if and only if $$\log(a_n)/n \to 0$$

The latter requires that $$\log(a_n)\in o(n)$$ which is satisfied by $$a_n=n^{k}$$, for any $$k \in \mathbb{R}$$ which is less restrictive that convergence for the corresponding series.

CB