A) \[\frac{e+1}{e-1}\]
B) \[\frac{e-1}{e+1}\]
C) \[\frac{{{e}^{2}}+1}{{{e}^{2}}-1}\]
D) \[\frac{{{e}^{2}}-1}{{{e}^{2}}+1}\]
Correct Answer: B
Solution :
\[\frac{\frac{1}{2\ !}+\frac{1}{4\ !}+\frac{1}{6\ !}+......\infty }{1+\frac{1\ }{3\ !}+\frac{1}{5\ !}+\frac{1}{7\ !}+.......\infty }\] \[=\,\frac{2\,\,\left[ \frac{1}{2\,!}+\frac{1}{4\,!}+\frac{1}{6\,!}+....\infty \right]}{2\,\,\,\left[ 1+\frac{1}{3\,!}+\frac{1}{5\,!}+\frac{1}{7\,!}+....\infty \right]}\] \[=\frac{(e+{{e}^{-1}})-2}{(e-{{e}^{-1}})}=\frac{e+\frac{1}{e}-2}{e-\frac{1}{e}}=\frac{{{e}^{2}}+1-2e}{{{e}^{2}}-1}\] \[=\frac{{{(e-1)}^{2}}}{(e-1)(e+1)}=\frac{e-1}{e+1}\].
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