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Give an example of a convergent series

Alexmahone

Active member
Jan 26, 2012
268
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
Jan 26, 2012
890
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