Suppose we have a non-negative function $f$. Suppose we have that the series $$ \sum_{n = 2}^{\infty} f(n) $$ converges. I was wondering does it then follow that $$ \sum_{n=2}^{\infty}f(n) \log n $$ converge as well? My guess is that there should be a counterexample for this but I have not been able to construct one. Any comments appreciated!
2026-04-03 17:55:25.1775238925
Convergence of a series with the summands changed by a factor of log?
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You can take $f(n)=\dfrac1{\log^2(n)}$, because$$\sum_{n=2}^\infty\frac1{n\log^2(n)}$$converges, but$$\sum_{n=2}^\infty\frac1{n\log(n)}$$diverges.