Find an example where $a_n\ge0$ and $\limsup na_n =\infty$ but $\sum^\infty_{k=0}a_n$ converges?

Asked Dec 13 '22 at 19:59

Active Dec 13 '22 at 19:59

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Find an example where $a_n\ge0$ and $\limsup na_n =\infty$ but $\sum^\infty_{k=0}a_n$ converges.

I have tried making the $a_n$ oscillate to $0$, but still can't see how to get $\limsup na_n =\infty$ because it seems like this requires $a_n$ to tend toward $0$ slower than $n$ to $\infty$.

asked Dec 13 '22 at 19:59

jenny9

See for example https://math.stackexchange.com/a/1825555/42969 – Martin R Dec 13 '22 at 20:02
The $\varlimsup$ allows you to get fancy from time to time in order to reach the infinite limit, while the rest of the sequence is well behaved and has a finite sum. – zwim Dec 13 '22 at 20:10

Find an example where $a_n\ge0$ and $\limsup na_n =\infty$ but $\sum^\infty_{k=0}a_n$ converges?

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