find a sequence with the following properties

Question

Does there exist a sequence $a_n$ with the following properties?

$a_{n+1} ≤ a_{n}$ for every $n$
$\displaystyle ∑a_n$ converges
$\displaystyle ∑ln(n)a_n$ diverges

If such a sequence exists please provide it, otherwise please give a proof (or a link to a proof) that such a sequence does not exist.

Thanks

Answer 1

You can take an ad hoc Bertrand's series, say $$\sum_{n>1}\frac 1{n\log^2n}.$$ The denominators decrease to $0$, it is well-known to converge (by the integral test, for instance). However $$\sum_{n>1}\log n \,a_n=\sum_{n>1}\frac 1{n\log n}$$ is known to diverge, by the same test.