anonymous
  • anonymous
How to find the sum of k^2/k! from 0 to infinity?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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amistre64
  • 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
anonymous
  • anonymous
Can i use the fact that the sum of 1/k! from 0 to infinity is e-1
amistre64
  • amistre64
i cant say for sure .... you can give it a shot and then verify it with the wolf

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amistre64
  • amistre64
you can try taking the partial sums up to some arbitrary point, and then averaging the integrals of the remainder
amistre64
  • amistre64
but i got no idea how to intgegrate a factorial at the moment
amistre64
  • 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
sirm3d
  • sirm3d
\[\large \frac{ k^2 }{ k! }=\frac{k(k-1+1)}{k!}=\frac{k(k-1)}{k!}+\frac{k}{k!}=\frac{1}{(k-2)!}+\frac{1}{(k-1)!}\]
sirm3d
  • sirm3d
so the answer is 2e.
amistre64
  • amistre64
good job :)
sirm3d
  • sirm3d
^^

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