Coefficient of xn in 1+x1!+(1+x)22!+(1+x)33!+……=

1+x1+(1+x)22+(1+x)33+…… =e(1+x)-1 =e ex-1 =e1+x+x22+x33+...+xnn+1... -1xn coef=e1n

Coefficient of xn in 1+x1!+(1+x)22!+(1+x)33!+……=

1+x1+(1+x)22+(1+x)33+…… =e(1+x)-1 =e ex-1 =e1+x+x22+x33+...+xnn+1... -1xn coef=e1n

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Coefficient of $x^{n}$ in $\frac{1 + x}{1!} + \frac{{(1 + x)}^{2}}{2!} + \frac{{(1 + x)}^{3}}{3!} + \dots \dots =$

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$\frac{en}{n!}$

$\frac{1}{e \cdot n!}$

$\frac{e}{n!}$

$\frac{e}{n!} - 1$

answer is A.

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Detailed Solution

$\begin{array}{rcl} \frac{1 + x}{1} + \frac{{(1 + x)}^{2}}{2} + \frac{{(1 + x)}^{3}}{3} + \dots \dots \\ = & e^{(1 + x)} - 1 \\ = & e e^{x} - 1 \\ = & e (1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . . + \frac{x^{n}}{n} + 1 . . .) - 1 \end{array}$
$\begin{array}{rcl} x^{n} coef & = & e (\frac{1}{n}) \end{array}$