A) \[\frac{({{e}^{2}}-1)}{2}\]
B) \[\frac{{{(e-1)}^{2}}}{2e}\]
C) \[\frac{({{e}^{2}}-1)}{2e}\]
D) \[\frac{({{e}^{2}}-2)}{e}\]
Correct Answer: B
Solution :
We know that \[e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+....\infty \] ?(i) \[{{e}^{-1}}=1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-....\infty \] ?.(ii) On adding Eqs. (i) and (ii) \[e+{{e}^{-1}}=2+\frac{2}{2!}+\frac{2}{4!}+.....\infty \] \[\Rightarrow \] \[\frac{{{e}^{2}}+1}{e}-2=\frac{2}{2!}+\frac{2}{4!}+....\infty \] \[\Rightarrow \] \[\frac{{{e}^{2}}+1-2e}{e}=2\left[ \frac{1}{2!}+\frac{1}{4!}+....\infty \right]\] \[\Rightarrow \] \[\frac{{{(e-1)}^{2}}}{2e}=\frac{1}{2!}+\frac{1}{4!}+....\infty \]
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