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### Evaluate $\sum_{k=1}^\infty \frac{k^2}{(k-1)!}$. [duplicate]

Evaluate $\sum_{k=1}^\infty \frac{k^2}{(k-1)!}$ I sense the answer has some connection with $e$, but I don't know how it is. Please help. Thank you.
3k views

### Series $\frac{k^2}{k!}$ with $k=1$ to $\infty$ [duplicate]

A practice Math Subject GRE asked me to compute $\sum_{k=1}^\infty \frac{k^2}{k!}$. The sum is equal to $2e$, but I wasn't able to figure this out using Maclarin series or discrete PDFs. What's the ...
254 views

### Calculate sum of series $\sum \frac{n^2}{n!}$ [duplicate]

I have to calculate sum of series $\sum \frac{n^2}{n!}$. I know that $\sum \frac{1}{n!}=e$ but I dont know how can I use that fact here..
250 views

### Value of $\sum\limits_{n= 0}^\infty \frac{n²}{n!}$ [duplicate]

How to compute the value of $\sum\limits_{n= 0}^\infty \frac{n^2}{n!}$ ? I started with the ratio test which told me that it converges but I don't know to what value it converges. I realized I only ...
198 views

### Show $\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$ [duplicate]

I stumpled upon the equation $$\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$$ and was just curious how to deduce the right hand side of the eqution - which identities could be of use here? Trying ...
119 views

### How do I prove that: $\sum_{i=0}^{\infty} \frac{i^2}{i!}=2e$ [duplicate]

I've seen in Wolfram Alpha that $$\sum_{i=0}^{\infty} \frac{i^2}{i!}=2e$$ but I have no idea how to prove that. Can anyone help me? Thanks.
### What's the sum of the series $\sum\limits_{n\geq 0}\frac{n^x}{n!}$ with $x$ a positive real number?
By the ratio test the series $$\sum_{n\ge0}\frac{n^x}{n!}$$ is convergent, but I know no method to evaluate it. Since it's a convergent series then my question here is: Is there a closed form ...