# Thread: Convergence, Divergence of a series

1. ## Convergence, Divergence of a series

For the following series, determine its convergence or divergence. If it converges, find its sum:

a) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{n^2-1}$ converges to $\displaystyle 0$

b) $\displaystyle \sum_{n=1}^{\infty} \frac{n+1}{2n-1}$ diverges

c) $\displaystyle \sum_{n=2}^{\infty} \frac{n}{ln (n)}$ diverges

If some of the above are incorrect, could you please help me fix it?!

2. Originally Posted by qzno

For the following series, determine its convergence or divergence. If it converges, find its sum:

a) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{n^2-1}$ converges to $\displaystyle 0$

b) $\displaystyle \sum_{n=1}^{\infty} \frac{n+1}{2n-1}$ diverges

c) $\displaystyle \sum_{n=2}^{\infty} \frac{n}{ln (n)}$ diverges

If some of the above are incorrect, could you please help me fix it?!
(b) and (c) are ok ... the sum of (a) is not 0.

$\displaystyle \sum_{n=2}^{\infty} \frac{1}{n^2-1} = \frac{1}{2} \sum_{n=2}^{\infty} \frac{1}{n-1} - \frac{1}{n+1} = \frac{3}{4}$