Question

Find sum of series $\frac{1}{\mathrm{1.3.5}}+\frac{1}{\mathrm{3.5.7}}+\frac{1}{\mathrm{5.7.9}}+........$

Answer 1

$\frac{1}{\mathrm{1.3.5}}+\frac{1}{\mathrm{3.5.7}}+......................$

$s=\frac{1}{4}\left(\frac{4}{\mathrm{1.3.5}}+\frac{4}{\mathrm{3.5.7}}+.........+\frac{4}{\left(2n-1\right)\left(2n+1\right)\left(2n+3\right)}\right)$

$s=\frac{1}{4}\left(\frac{5-1}{\mathrm{1.3.5}}+\frac{7-3}{\mathrm{3.5.7}}+...........+\frac{\left(2n+3\right)-\left(2n-1\right)}{\left(2n-1\right)\left(2n+1\right)\left(2n+3\right)}\right)$

$s=\frac{1}{4}\left[\left(\frac{1}{1.3}-\frac{1}{3.5}\right)+\left(\frac{1}{3.5}-\frac{1}{5.7}\right)+..........+\left(\frac{1}{\left(2n-1\right)\left(2n+1\right)}-\frac{1}{\left(2n+1\right)\left(2n+3\right)}\right)\right]$

$s=\frac{1}{4}\left(\frac{1}{3}-\frac{1}{\left(2n+1\right)\left(2n+3\right)}\right)$

$s=\frac{1}{4}\left(\frac{1}{3}-\frac{1}{\left(2n+1\right)\left(2n+3\right)}\right)$