Infinite summation 21

Question

Find $$\frac{1}{3}+\frac{1}{10}+\frac{1}{21}+\frac{1}{45}+....\infty$$ I tried to rewrite it as $$S=\frac{1}{1\times3}+\frac{1}{2\times5}+\frac{1}{3\times7}+\frac{1}{5\times9}+.....$$

To identify the pattern but i could not go on

Answer 1

This is $\sum_{n=1}^\infty\frac1{n(2n+1)}$. This series converges. Just use the comparison test with the series $\sum_{n=1}^\infty\frac1{n^2}$.