1. ## test for convergence

show that the sum of 1-cos(pie/n) from n = 1 to infinity converges or diverges

second question:
show that the sum of e^-n/(n+1) from n =1 to infinity converges or diverges
i think it converges. is it okay if i use the comparison test by comparing series to another series and can i use the integral test to see if series being compared converges?

2. Doesn't the second one diverge by Nth term/TFD or am I mistaken?

4. #1: We're going to make use of the fact that for all $x \in (0, \pi)$, we have: $0 < 1 - \cos x < \tfrac{1}{2}x^2$. To prove this, consider $f(x) = 1 - \cos x - \tfrac{1}{2}x^2$ and prove that it is a decreasing function. Thus, $f(x) < f(0) = 0 \ \Leftrightarrow \ 1 - \cos x < \tfrac{1}{2}x^2$

So now we have the inequality: $0 < 1 - \cos \tfrac{\pi}{n} < \frac{\pi^2}{2n^2}$ since $\frac{1}{n} \in (0, \pi)$ for all $n$.

Now use comparison.

#2: Use the divergence test. If $\lim_{n \to \infty} a_n$ is not equal to 0, then the series $\sum a_n$ diverges.

5. ## thanks

thanks

6. how'd u get 1/2x^2

7. It's simply a fact: $0 < 1 - \cos x < \frac{x^2}{2}$ for $0 < x < \pi$