Finding the summation of these two Infinite Series

Question

I find the sequence of partial sums, but I cannot find the general term. I know once I find the general term of the sequence of partial sums, I take the limit of that and that number is the sum of the series. I just cannot find a pattern in either of these.

Your help would be greatly appreciated.

Answer 1

Note that

• $\sum \frac{2}{n^2+4n+3}=\sum \frac{1}{n+1}-\sum\frac{1}{n+3}$ and many terms cancel out
• $\sum \frac{1+2^n}{3^n}=\sum \left(\frac13\right)^n+\sum \left(\frac23\right)^n$ and refer to geometric series

Answer 2

To elaborate on the comment hint, write the first sum as $$ \sum \left(\frac{A}{n+B} - \frac{C}{n+D}\right) $$ and apply telescoping series. For the second one note that $$ \frac{1+2^n}{3^n} = \frac{1}{3^n} + \frac{2^n}{3^n} = (1/3)^n + (2/3)^n $$