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Is there a function $f : \mathbb{N} \to \mathbb{R}^+$ in $o(1/n)$ s.t. $\Sigma_{i=0}^\infty f(n)$ diverges?

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up vote 2 down vote accepted

Yes. let $g(n)=\ln\ln n$. Then $g(n+1)-g(n)\approx g'(n)=\frac 1n\cdot\frac1{\ln n}\in o(\frac1n)$. Thus letting $f(n)=g(n+1)-g(n)$, you have your example

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Try $$\sum_{i=2}^\infty \frac{1}{n \log(n) }$$

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And use the integral test. –  Brian M. Scott Sep 19 '12 at 6:36
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