# $\lim_{N\to \infty} \sum_1^N\left(1-\frac{n}{N+1}\right)a_n$ for any positive sequence $\{a_n\}$

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?

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 $$\liminf_{N\to \infty} \sum_1^N\left(1-\frac{n}{N+1}\right)a_n\ge\lim_{N\to \infty} \sum_1^K\left(1-\frac{n}{N+1}\right)a_n=\sum_1^K a_n.$$