To find sum of infinite series.

Question

How to find the sum of the following infinite series?

$$\frac1{2 \cdot 3 \cdot 4}+\frac1{4 \cdot 5 \cdot 6}+\frac1{6 \cdot 7 \cdot 8}+\cdots$$

I can write its partial sum but am unable to proceed.

Answer 1

$$\frac1{(n-1)n(n+1)}=\frac12\left(\frac1{n-1}-\frac2n+\frac1{n+1}\right)$$ so your sum is $$\frac12\left(\frac12-\frac23+\frac24-\frac25+\frac26-\frac27+\cdots\right).$$ This is very similar to $$\frac11-\frac12+\frac13-\frac14+\cdots.$$