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series diverges, a_n/(1+na_n)

Source: VJIMC 2004 2.3

July 12, 2021
ConvergenceSequenceslimitsreal analysis

Problem Statement

Let ∑n=1∞an\sum_{n=1}^\infty a_n be a divergent series with positive nonincreasing terms. Prove that the series ∑n=1∞an1+nan\sum_{n=1}^\infty\frac{a_n}{1+na_n}diverges.
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