$\sum_{n=2}^\infty\frac1{(n\log n)^{2-\sin n}}$ diverges?

Asked May 11 '23 at 05:34

Active May 13 '23 at 06:16

Viewed 101 times

I know that $$\sum_{n=1}^\infty\frac1{n^{2-\sin n}}$$ diverges.

But how about $$\sum_{n=2}^\infty\frac1{(n\log n)^{2-\sin n}}\,?$$

I think it would be very interesting if it converges.

Since ${n\log n}$ and power operation $n^{2-\sin n}$ both are less dense than $n$ itself, on the number line, but still not dense enough to make the sum convergent.

edited May 13 '23 at 06:16

metamorphy

39,111

asked May 11 '23 at 05:34

mymemeisJo

1

Hello, the series in the title and in the question are different. Can you clarify which one you are interested in? – Taladris May 11 '23 at 05:37
Sorry for the mistake. I edited. By the way, I also interested in $\sum_{n=2}^{\infty}\frac{1}{n{(log n)}^{2-\sin(n)}}$ not only $\sum_{n=2}^{\infty}\frac{1}{{(n\log n)}^{2-\sin(n)}}$ – mymemeisJo May 11 '23 at 05:39
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"I know that $\sum_{n=1}^{\infty}\frac{1}{{n}^{2-\sin(n)}}$ diverges." : Please explain how you know this. – user2661923 May 11 '23 at 05:48
Sorry for that. By the reference: https://math.stackexchange.com/questions/2626391/convergence-or-divergence-of-sum-n-1-infty-frac1n2-cos-n – mymemeisJo May 11 '23 at 05:50
Please also present your attempts (even if they failed) to adapt that proof to your series. Quick beginner guide for asking a well-received question – Anne Bauval May 11 '23 at 05:51
Sorry for that too. I tried several test but I need a little time to organize them and edit. I will edit it as fast as I can. – mymemeisJo May 11 '23 at 05:54
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Don't waste your time on the usual tests, which are useless here. Again: please present how you tried to adapt that proof (of divergence of $\sum_{n=1}^{\infty}\frac{1}{{n}^{2-\sin(n)}}$) to your series. – Anne Bauval May 11 '23 at 05:56

$\sum_{n=2}^\infty\frac1{(n\log n)^{2-\sin n}}$ diverges?

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