Question

The sum of the series $1+\frac{1}{5}+\frac{1.3}{5.10}+\frac{\mathrm{1.3.5}}{\mathrm{5.10.15}}+....$ is equal to

• A. $\frac{1}{\sqrt{5}}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\sqrt{\frac{5}{3}}$
• D. $\sqrt{5}$

Answer 1

The correct option is C $\sqrt{\frac{5}{3}}$

$S=1+\frac{1}{5}+\frac{1.3}{5.10}+\frac{\mathrm{1.3.5}}{\mathrm{5.10.15}}+....(1+x{)}^n=1+\frac{nx}{1!}+\frac{n(n-1)}{2!}{x}^2+\frac{n(n-1)(n-2)}{3!}{x}^3+..\Rightarrow nx=\frac{1}{5}and\frac{n(n-1)x^2}{2!}=\frac{1.3}{5.10}\Rightarrow n=-\frac{1}{2}andx=\frac{-2}{5}\therefore S=(1-\frac{2}{5}{)}^{-\frac{1}{2}}=(\frac{3}{5}{)}^{-\frac{1}{2}}=\sqrt{\frac{5}{3}}$