5
$\begingroup$

Can you help me find

$\lim_{n \to \infty}\frac1{n^3}\sum_{k=0}^{n-1}k^2e^{-\frac{k}{n}} \ \ ?$

Using Riemann sums, I found it is equal to $2-\frac3{e}$. Is that correct ? Thank you in advance.

  • 0
    The devil is often in the details, so it's possible for your work to be subtly incorrect even if you got the right answer. It would be more beneficial to you, and quicker for the answerers, if you include the entire work rather than just the final answer.2011-12-31

1 Answers 1

10

This is just $ \lim_{n \to \infty} \frac 1{n^3} \sum_{k=0}^{n-1} k^2 e^{-k/n} = \lim_{n \to \infty} \sum_{k=0}^{n-1} \left( \frac {k+1}n - \frac kn \right) \left( \frac kn \right)^2 e^{- \left( \frac kn \right) } = \int_0^1 x^2 e^{-x} \, dx. $ Do that and you got your answer (integrate by parts twice). Looks like you already did so I won't detail more.

Hope that helps,

  • 0
    Well I didn't even bother computing the integral... it looks quite elementary. If OP wanted the answer he could've just computed this integral in Wolfram Alpha if he didn't trust himself. =)2011-12-31