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convergence of sequence sum with 1/(na_n^2)

Source: VTRMC 2010 P7

May 17, 2021
Sequencesreal analysisSumSummation

Problem Statement

Let ∑n=1∞an\sum_{n=1}^\infty a_n be a convergent series of positive terms (so ai>0a_i>0 for all ii) and set bn=1nan2b_n=\frac1{na_n^2} for n≥1n\ge1. Prove that ∑n=1∞nb1+b2+…+bn\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n} is convergent.
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