Let $\sum_{n=1}^\infty a_n$ be an arbitrary absolutely convergent series. Can we show that $$ \exists C>0 : a_n \le \frac{C}{n}, \quad \forall n \in \mathbb{N}. $$
2026-04-26 02:48:29.1777171709
Every absolutely convergent series can be bounded by the harmonic series
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No. Define $a_n$ by saying that $a_n=0$ if $n$ is not a perfect cube, while $a_n=1/j^2$ if $n=j^3$. Then $\sum a_n=\sum1/j^2<\infty$, but if $n=j^3$ then $na_n=j$, so $na_n$ is not bounded.