Question

Find the sum of n terms of the series $\frac{1}{\mathrm{1.2.3.4}}+\frac{1}{\mathrm{2.3.4.5}}+\frac{1}{\mathrm{3.4.5.6}}+....$

Answer 1

The $n^{th}$ term is $\frac{1}{n(n+1)(n+2)(n+3)};$
hence, by the rule, we have
$S_n=C-\frac{1}{3(n+1)(n+2)(n+3)}$.
Put $n=1$, then $\frac{1}{\mathrm{1.2.3.4}}=C-\frac{1}{\mathrm{3.2.3.4}};$ hence $C=\frac{1}{19};$
$\therefore S_n=\frac{1}{18}-\frac{1}{3(n+1)(n+2)(n+3)}$.
By making n idefinitely great, we obtain $S_{\infty }=\frac{1}{18}$.