# Showing that $\sum k^{2}/k!=e^2$ [duplicate]

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I'm having trouble seeing why $\sum_{k=1} k^{2}/k!=e^2$. The ratio test says that it converges absolutely. Pardon my ignorance, but are there any techniques to show this?

I thought about expanding $e^{2}= e \cdot e =(\sum_{k=0} \frac{1}{k!})(\sum_{k=0} \frac{1}{k!})$ but so far that hasn't helped me much.

## marked as duplicate by Clement C., user135520, Community♦Sep 5 '16 at 5:57

• See the Taylor expansion of $e^x$ (it is google-able) – Simply Beautiful Art Sep 4 '16 at 23:00
• @SimpleArt Given the question, it looks like the OP is aware of that. – Clement C. Sep 4 '16 at 23:05
• Sorry guys for the mistake and the duplicate, I voted to close. – user135520 Sep 5 '16 at 5:53

You may have misread your source, this is not equal to $e^2$, but to $2e$. Unrolling the expression, \begin{align} \sum_{k=0}^\infty \frac{k^2}{k!} &= \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!} \\ & = \sum_{k=1}^\infty \frac{k}{k!} +e^1 = \sum_{k=1}^\infty \frac{1}{(k-1)!} +e^1 = \sum_{k=0}^\infty \frac{1}{k!} +e^1 = e^1 +e^1 \\ &= 2e \end{align}

• All too easy. +1. – Mark Viola Sep 5 '16 at 2:13

\begin{align*} \sum_{k=0}^\infty \frac{k^2}{k!}&=\sum_{k=1}^\infty \frac{k}{(k-1)!} =\sum_{k=1}^\infty \frac{k-1}{(k-1)!}+\sum_{k=1}^\infty \frac{1}{(k-1)!} \\ &= \sum_{k=2}^\infty \frac{k-1}{(k-1)!}+\sum_{k=0}^\infty \frac{1}{k!}=\sum_{k=2}^\infty \frac{1}{(k-2)!}+e =\sum_{k=0}^\infty \frac{1}{k!}+e =2e \end{align*}

$e^x= \sum \frac{x^k}{k!}$

$e^x= (e^x)'= \sum \frac{x^{(k-1)}}{(k-1)!}$

$x e^x = \sum \frac{x^k}{(k-1)!}$

$(x e^x)' = (x+1)e^x = \sum \frac{k x^{(k-1)}}{(k-1)!} = \sum \frac{k^2 x^{(k-1)}}{k!}$

so $2e = \sum \frac{k^2}{k!}$

No. Actually,

$\sum_{k=0} 2^{k}/k!=e^2$

• The fact that another series has sum $e^2$ does not mean that this one cannot sum to $e^2$. (In this case, it does not, but the argument is not a valid one.) – Clement C. Sep 4 '16 at 23:04
• @ClementC. sure. But many people throw random slop here without proofreading. I might have worried, but you and another found the actual value of the series as typed. – Will Jagy Sep 4 '16 at 23:10