I read that an alternating series [tex] \Sigma (-1)^n a_n [/tex] converges if "and only if" the sequence [tex]a_n [/tex] is both monotonous and converges to zero.(adsbygoogle = window.adsbygoogle || []).push({});

I tried with this series:

[tex] \Sigma_{n=1}^{\infty} (-1)^n | \frac{1}{n^2} \sin(n)| [/tex]

in the wolfram alpha and seems to converge to -0.61..., even if [tex] a_n = |\frac{1}{n^2} \sin(n)| [/tex] is not monotonous decreasing.

What am I doing wrong? Is the monotone condition necesary for this test, but the fact that a_n is not monotonous does not guarantee if the series converges or not?

Thanks.

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# Alternating series (Leibniz criterion)

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