## anonymous 3 years ago How to find the sum of k^2/k! from 0 to infinity?

1. amistre64

if there was a pattern we could find from the partial sums, that could be useful; but its still a shot in the dark

2. anonymous

Can i use the fact that the sum of 1/k! from 0 to infinity is e-1

3. amistre64

i cant say for sure .... you can give it a shot and then verify it with the wolf

4. amistre64

you can try taking the partial sums up to some arbitrary point, and then averaging the integrals of the remainder

5. amistre64

but i got no idea how to intgegrate a factorial at the moment

6. amistre64

$e=\frac1{0!}+\frac1{1!}+\frac1{2!}+\frac1{3!}+\frac1{4!}+\frac1{5!}+...$ $k^2e=k^2\left(\frac1{0!}+\frac1{1!}+\frac1{2!}+\frac1{3!}+\frac1{4!}+\frac1{5!}+...\right)$ pfft, theres prolly something wrong with my idea on that one

7. anonymous

$\large \frac{ k^2 }{ k! }=\frac{k(k-1+1)}{k!}=\frac{k(k-1)}{k!}+\frac{k}{k!}=\frac{1}{(k-2)!}+\frac{1}{(k-1)!}$

8. anonymous

9. amistre64

good job :)

10. anonymous

^^