Example of divergent series whose associated "squared" series converges

Asked Jun 20 '17 at 14:31

Active Jun 20 '17 at 14:31

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I am trying to find a non-trivial example of a series with positive terms $a_n$ that satisfies the following two conditions:

$$ \sum_{n = 1}^{+\infty} a_n = +\infty \qquad \text{and} \qquad \sum_{n = 1}^{+\infty} a_n^2 < +\infty. $$

The obvious example is of course $a_n = \frac{1}{n}$ for all $n \in \mathbb{N}\setminus \{0\}$, but could you help me finding another one? Thank you very much for your answers.

asked Jun 20 '17 at 14:31

Lightspark

3

Slightly less trivial: $a_n = 1/n^p$ for any $p \in (1/2,1]$ – Jun 20 '17 at 14:34
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You might find the divergence of prime reciprocals interesting. It supplies a "modern" proof of the infinitude of primes. Convergence of the squared prime reciprocals is also previously discussed. – hardmath Jun 20 '17 at 14:45
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Another class of less trivial $n^{a+p}\sin(\frac{1}{n^{2p+a}})$ for $p\in(1/2,1]$ and any $a$. – kingW3 Jun 20 '17 at 14:47

Example of divergent series whose associated "squared" series converges

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