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Is the following series convergent or not and why? $\sum_{n = 1}^{\infty} \cot(\pi/2 - 1/n)$.

I don't know why I cannot get this, but I was expecting either $\cot(\pi/2 - 1/n) < x^{2}$ or $\cot(\pi/2 - 1/n) > x$ near $x = 0$. I cannot show neither of them at this moment; hence the question. Thanks in advance.

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1 Answer 1

up vote 5 down vote accepted

We have

$$\cot\left(\frac\pi2-\frac1n\right)=\tan\left(\frac1n\right)\sim_\infty\frac1n$$ so the given series is divergent by comparison with the harmonic series $\sum_n\frac1n$.

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I am such a fool... Ain't I? Thank you! – user123454321 Jul 6 '14 at 2:42
You're welcome:-) – user63181 Jul 6 '14 at 2:45

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