The sum 1+1+3 2! +1+3+5 3! +1+3+5+7 4! + upto terms, is equal to - Vidyadip

MCQ

The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to

A
$6 e$
B
$4 e$
C
$3 e$
D
$2 e$

✓

Answer

D. $2 e$
$\mathrm{S}=1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\ldots$
$=\sum_{r=1}^{\infty} \frac{r^{2}}{r!}$
$=\sum_{\mathrm{r}=1}^{\infty} \frac{(\mathrm{r}-1+1)}{(\mathrm{r}-1)!}=\sum_{\mathrm{r}=2}^{\infty} \frac{1}{(\mathrm{r}-2)!}+\sum_{\mathrm{r}=1}^{\infty} \frac{1}{(\mathrm{r}-1)!}$
$=2 \mathrm{e}$

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