A calculus problem by Bharath Sriraam

Calculus Level 4

Compute the value of the following infinite sum:

$\displaystyle \sum _{ r=1 }^{ \infty }{ \frac { { r }^{ 8 } }{ r! } }$

NOTE

$e =\displaystyle \lim_{x\rightarrow \infty} \left(1+\dfrac{1}{x}\right)^{x}$ is the Euler's number.

BONUS: Find $\displaystyle \sum _{ r=1 }^{ \infty }{ \frac { { r }^{ n } }{ r! } } =?$

4140e 9690e 5250e 2016e

1 solution

First Last
Jul 7, 2017

One could use the following for smaller values than 8:

$\displaystyle\sum_{n=1}^\infty\frac{x^n}{n!}+1=e^x\quad x\frac{d}{dx}\bigg(\sum_{n=1}^\infty\frac{x^n}{n!}+1\bigg)=\sum_{n=1}^\infty\frac{nx^n}{n!}=xe^x$

And continue differentiating indefinitely until a desired power of n in the numerator, but this is tedious. To generalize, the Bell numbers are defined as $B_n=\frac1{e}\sum_{k=0}^\infty\frac{k^n}{k!}$

We need only $B_8=4140$ and Wikipedia gives a nice way of calculating them recursively.

A calculus problem by Bharath Sriraam

1 solution

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