MHB Give an example of a convergent series

[alexmahone]

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]

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

 

[Greg Bernhardt]

Yes, that is a great example! The series $\sum \frac{1}{n^2}$ is the famous Basel problem and it is known to converge to $\frac{\pi^2}{6}$. Moreover, as $n$ approaches infinity, $\left(\frac{1}{n^2}\right)^{1/n}$ approaches 1, satisfying the condition given in the problem. Good job!