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If I have that $|\sum |a_n| - |a_n|^2|$ is bounded and know that $\sum|a_n|^2$ converges can I conclude anything about $\sum a_n$?

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up vote 3 down vote accepted

By the triangular inequality, $$ \sum_n|a_n|\leqslant\left|\sum_n|a_n|-|a_n|^2\right|+\sum_n|a_n|^2 $$ is bounded, hence $\sum\limits_na_n$ converges absolutely. In particular, $\sum\limits_na_n$ converges.

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Since $\sum_{n\geq 1}|a_n|^2$ is convergent, in particular $a_n\to 0$ so for $n\geq n_0$ we have $|a_n|\leq 1$. We get that $$\sum_{n\geq n_0}|a_n|= \sum_{n\geq n_0}\left(|a_n|-|a_n|^2\right)+\sum_{n\geq n_0}|a_n|^2\leq \sup_{N\in \mathbb N}\left|\sum_{j=1}^N|a_j|-|a_j|^2\right|+\left|\sum_{j=1}^{n_0}|a_j|-|a_j|^2\right|+\sum_{k\geq 1}|a_k|^2<\infty,$$ so the series $\sum_n|a_n|$ is convergent.

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