Question

Find the sum of the following series $\frac{1}{\mathrm{1.2.3}}+\frac{1}{\mathrm{3.4.5}}+\frac{1}{\mathrm{5.6.7}}+....$

Answer 1

$\Rightarrow u_n=\frac{1}{(2n-1)}.\frac{1}{2n}.\frac{1}{(2n+1)}$

$=\frac{1}{2}\{\frac{1}{2n-1}-\frac{2}{n}+\frac{1}{2n+1}\}$

$2S=(\frac{1}{1}-\frac{2}{2}+\frac{1}{3})+(\frac{1}{3}-\frac{2}{4}+\frac{1}{5})+(\frac{1}{5}-\frac{2}{6}+\frac{1}{7})+......$

$1+2S=2(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+....)$

$1+2S=2{\log }_{e}2$

$2S={\log }_{e}4-{\log }_{e}e$

$S=\frac{1}{2}{\log }_e\frac{4}{e}$