Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 $\sum_n\frac1n$.

share|cite|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.