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if sum(a_n/n) converges then sum(a_n)/n converges

Source: VJIMC 2006 1.2

June 28, 2021
limitsConvergence

Problem Statement

Suppose that (an)(a_n) is a sequence of real numbers such that the series ∑n=1∞ann\sum_{n=1}^\infty\frac{a_n}nis convergent. Show that the sequence bn=1n∑j=1najb_n=\frac1n\sum^n_{j=1}a_jis convergent and find its limit.
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