The value of the series \frac{2}{3!}+\frac{4}{5!}+\frac{6}{7!}+\cdots is

NIMCET 2024 Paper

Section: Mathematics

Question : 50 of 120

Marks: +1, -0

The value of the series

\;{2}/{3!}+\;{4}/{5!}+\;{6}/{7!}+⋯

Solution:

n^{\;\text "th "\;}

term is given by

\;{2 n}/{(2 n+1)!}=\;{2 n+1-1}/{(2 n+1)!}=\;{1}/{2 n!}-\;{1}/{(2 n+1)!}

= (\;{e+e^{-1}}/{2}-1) - (\;{e-e^{-1}}/{2}-1) =e^{-1}

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