The sum of the series $1+\frac{1}{4.2 !}+\frac{1}{16.4 !}+\frac{1}{64.6 !}+\ldots \ldots \ldots$ ad inf. is

The sum of the series $1+\frac{1}{4.2 !}+\frac{1}{16.4 !}+\frac{1}{64.6 !}+\ldots \ldots \ldots$ ad inf. is
  1. $\frac{\mathrm{e}-1}{\sqrt{\mathrm{e}}}$
  2. $\frac{\mathrm{e}+1}{\sqrt{\mathrm{e}}}$
  3. $\frac{\mathrm{e}-1}{2 \sqrt{\mathrm{e}}}$
  4. $\frac{e+1}{2 \sqrt{e}}$

Solution

$\frac{e^x+e^{-x}}{2}=1+\frac{x^2}{2 !}+\frac{x^4}{4 !}+\frac{x^6}{6 !}+\ldots \ldots$ putting $x=1 / 2$ we get $\frac{e+1}{2 \sqrt{e}}$

Asked in: JEE Main 2005

Practice more Sequences and Series questions on Aicharya