1+14.2!+116.4!+164.6!+……=

1+142+1164+1646+.... 1+1222+1244+1266+.... =e1/2+e-1/22=e+12e

1+14.2!+116.4!+164.6!+……=

1+142+1164+1646+.... 1+1222+1244+1266+.... =e1/2+e-1/22=e+12e

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$1 + \frac{1}{4.2!} + \frac{1}{16.4!} + \frac{1}{64.6!} + \dots \dots =$

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$\frac{e + 1}{\sqrt{e}}$

$\frac{e + 1}{2 \sqrt{e}}$

$\frac{e - 1}{\sqrt{e}}$

$\frac{e - 1}{2 \sqrt{e}}$

answer is D.

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Detailed Solution

$\begin{array}{rcl} 1 + \frac{1}{42} + \frac{1}{164} + \frac{1}{646} + . . . . \\ 1 + \frac{{(\frac{1}{2})}^{2}}{2} + \frac{{(\frac{1}{2})}^{4}}{4} + \frac{{(\frac{1}{2})}^{6}}{6} + . . . . \\ = & \frac{e^{1 / 2} + e^{- 1 / 2}}{2} = \frac{e + 1}{2 \sqrt{e}} \end{array}$