+54 Hello!
I have been trying to play around with this summation for a while, but was stuck at a certain place.
Problem: \(\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}.\)
My intake was to use the "Partial Fraction Decomposition method"
- \(\frac{2n+1}{n(n+1)(n+2)}=\frac{A}{n}+\frac{B}{n+1}+\frac{C}{n+2}\).
After some expanding, evaluating, and simplifying, I ended up with \(A=\frac{1}{2}, B=1, \) and \(C=\frac{-3}{2}\).
Therefore, I rewrote the series as: \(\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}=\sum_{n = 1}^\infty \frac{\frac{1}{2}}{n}+\frac{1}{n+1}+\frac{\frac{-3}{2}}{n+2}.\).
After this, I'm stuck, and I wonder if this series would telescope..Any help would be appreciated.
Thank you !
