Is it true that if $(a_n) \geq 0$ and $\displaystyle\sum_1^\infty a_n$ diverges, then $\displaystyle\sum_1^\infty a_n(1-r^n)$ diverges for all $r \in (0,1)$? I think it's true but I'm having a hard time proving it.
2026-04-02 17:49:59.1775152199
Question about divergent series
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Hint: If $\displaystyle\sum_1^\infty a_n(1-r^n)$ converges then $\displaystyle a_n = \frac{a_n(1-r^n)}{1-r^n} \to 0$ and hence $\displaystyle\sum_1^\infty a_n r^n$ also converges.