Convergence of a series

[elim]

Suppose \(\left\{b_{k}\right\}\) has the property that:
\(\sum\limits_{k=1}^n \frac{n-k}{n} |b_{k}| \leq M \quad (\forall n)\quad\) Show that \(\sum\limits_{k=1}^\infty |b_{k}| < \infty\)

[elim]

\(\sum_{k=1}^n |b_k| \le 2\sum_{k=1}^n \frac{2n-k}{2n}|b_k| \le 2\sum_{k=1}^{2n} \frac{2n-k}{2n} |b_k| \le 2M\)

[elim]

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