A) \[\frac{{{1}^{r}}}{1\,!}+\frac{{{2}^{r}}}{2\,!}+\frac{{{3}^{r}}}{3\,!}+.....\]
B) \[1+\frac{1}{1\,!}+\frac{1}{2\,!}+....+\frac{1}{r\,!}\]
C) \[\frac{1}{r\,!}\left[ \frac{{{1}^{r}}}{1\,!}+\frac{{{2}^{r}}}{2\,!}+\frac{{{3}^{r}}}{3\,!}+.... \right]\]
D) \[\frac{{{e}^{r}}}{r!}\]
Correct Answer: C
Solution :
Expansion of \[{{e}^{{{e}^{x}}}}\] \[=1+\frac{{{e}^{x}}}{1!}+\frac{{{e}^{2x}}}{2!}+\frac{{{e}^{3x}}}{3!}+\frac{{{e}^{4x}}}{4!}+....+\frac{{{e}^{rx}}}{r!}+....\] \[=1+\frac{1}{1!}\left[ 1+\frac{x}{1!}+\frac{{{x}^{2}}}{2!}+....+\frac{{{x}^{r}}}{r!}+..... \right]\]\[+\frac{1}{2!}\left[ 1+\frac{2x}{1!}+\frac{{{(2x)}^{2}}}{2!}+....+\frac{{{(2x)}^{r}}}{r!}+..... \right]\] + ??? Hence the coefficient of \[{{x}^{r}}=\frac{1}{r!}\left[ \frac{{{1}^{r}}}{1!}+\frac{{{2}^{r}}}{2!}+\frac{{{3}^{r}}}{3!}+\frac{{{4}^{r}}}{4!}+.... \right]\].
You need to login to perform this action.
You will be redirected in
3 sec
