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Point of convergence of a series

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Hello MHB.

I have been preparing for my subject GRE and I need help on the following problem.

Find $\displaystyle\sum_{k=1}^\infty \frac{k^2}{k!}$.

Using the ratio test we know that the series converges but how to we find what it converges to?
 

chisigma

Well-known member
Feb 13, 2012
1,704
Hello MHB.

I have been preparing for my subject GRE and I need help on the following problem.

Find $\displaystyle\sum_{k=1}^\infty \frac{k^2}{k!}$.

Using the ratio test we know that the series converges but how to we find what it converges to?
Is...

$\displaystyle \sum_{k=1}^{\infty} \frac{k^{2}}{k!} = \sum_{k=1}^{\infty} \frac{k}{(k-1)!} = \sum_{k=0}^{\infty} \frac{k+1}{k!} = \sum_{k=0}^{\infty} \frac{k}{k!} + \sum_{k=0}^{\infty} \frac{1}{k!} = 2\ e$

Kind regards

$\chi$ $\sigma$
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Is...

$\displaystyle \sum_{k=1}^{\infty} \frac{k^{2}}{k!} = \sum_{k=1}^{\infty} \frac{k}{(k-1)!} = \sum_{k=0}^{\infty} \frac{k+1}{k!} = \sum_{k=0}^{\infty} \frac{k}{k!} + \sum_{k=0}^{\infty} \frac{1}{k!} = 2\ e$

Kind regards

$\chi$ $\sigma$
Taught me a lot. Thanks. :)