Does there exist a convergent series $\sum _{n=1}^\infty a_n$ of positive terms such that $na_n$ does not converge to $0$ ?

Does there exist a convergent series $\sum _{n=1}^\infty a_n$ of positive terms such that $na_n$ does not converge to $0$ ? I only know that if such a series exists then the sequence $\{a_n\}$ cannot be decreasing . Please help . Thanks in advance

• hint : $\displaystyle\sum_{n \text{ is a square}} \frac{1}{n}$ Commented Sep 19, 2016 at 8:50

Consider $$a_n=\left\{\begin{array}{} \frac1n&\text{if n=2^k for k\in\mathbb{Z}}\\ \frac1{n^2}&\text{otherwise} \end{array}\right.$$

• Very good answer. The sequence has to be positive though, this can be done with a minor edit. Commented Oct 19, 2017 at 23:43
• @i707107: fixed
– robjohn
Commented Oct 20, 2017 at 0:24
• @i707107: Many times questions that are about non-negative numbers are asked about positive numbers. Often, the proof is more complicated, but no more instructive, if we are required to use positive numbers rather than non-negative numbers. In my edited answer, the $\frac1{n^2}$ draws attention away from the important part of the answer, but adds nothing of value other than to make all the terms positive.
– robjohn
Commented Oct 20, 2017 at 0:42

Yes, there exist such series. One cute example is $\sum_{n=1}^\infty a_n$ where: $$a_n = \begin{cases} 0, & \text{if the decimal expansion of n contains a 9} \\ \frac{1}{n}, & \text{otherwise}. \end{cases}$$

See for example this answer for more details.

(If you insist on positive terms, replace $0$ with something tending to $0$ quickly.)
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