1
$\begingroup$

How can I show that $\sum_{n=1}^\infty (\sin n -\sin(πn/2)) / n^2$ converges or diverges? I tried using the ratio test but it's complicated I'd say it converges to $0$ since $n^2$ is growing faster than the numerator.

$\endgroup$
1
$\begingroup$

Hint : $$\left| \frac{\sin n-\sin(\pi n/2)}{n^2}\right|\le \frac{2}{n^2}$$

and use the fact 'Every absolutely convergent series is convergent.'

$\endgroup$
  • $\begingroup$ I need to find a smaller series that will also converge in this case 1/n² and by the comparison the original series will converge right.. $\endgroup$ – mimi Oct 15 '13 at 1:58
  • $\begingroup$ You don't need a smaller series, you need a larger series, and tetori gave that to you. $\endgroup$ – Robert Israel Oct 15 '13 at 2:09
  • $\begingroup$ No. You want a larger series that converges. A smaller series will tell you that the series diverges if the smaller series diverges. $\endgroup$ – marty cohen Oct 15 '13 at 2:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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