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Let $\{a_n\}$ be a positive sequence. Suppose that
$\lim_{N\to \infty} \sum_1^N\left(1-\frac{n}{N+1}\right)a_n$ exists. Is the series $\sum a_n$ convergent?

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Given that $a_n\ge0$ this is trivial. If $\sum a_n$ does not converge then $\sum a_n=+\infty$, and it's obvious that that implies $\lim_{N\to \infty} \sum_1^N\left(1-\frac{n}{N+1}\right)a_n=\infty$: If $K$ is fixed then $$\liminf_{N\to \infty} \sum_1^N\left(1-\frac{n}{N+1}\right)a_n\ge\lim_{N\to \infty} \sum_1^K\left(1-\frac{n}{N+1}\right)a_n=\sum_1^K a_n.$$

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