How do you determine the convergence or divergence of Sigma sin(((2n-1)pi)/2) from [1,oo)?

Jan 16, 2017

Not convergent.

Explanation:

$\frac{\left(2 n - 1\right) \pi}{2} = n \pi - \frac{\pi}{2}$

so $\sin \left(n \pi - \frac{\pi}{2}\right) = - \cos \left(n \pi\right) = - {\left(- 1\right)}^{n}$

Finally

$\sum \sin \left(\frac{\left(2 n - 1\right) \pi}{2}\right) = - \sum {\left(- 1\right)}^{n}$ which is not convergent