Linked Questions

10
votes
5answers
436 views

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.
10
votes
4answers
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 ...
5
votes
3answers
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..
3
votes
4answers
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 ...
2
votes
4answers
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 ...
3
votes
3answers
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.
8
votes
4answers
144 views

Calculating the sum of $\sum\frac{n^2-2}{n!}$ [duplicate]

Calculating the sum of $\sum\frac{n^2-2}{n!}$ I want to calculate the sum of $\sum_{n=0}^{+\infty}\frac{n^2-2}{n!}$. This is what I have done so far: $$ \sum_{n=0}^{+\infty}\frac{n^2-2}{n!}=\sum_{...
2
votes
3answers
101 views

How to calculate $\sum_{n=1}^{\infty}\frac {n^2} {n!}$ [duplicate]

Summation of n²/n! where n takes the values from 1 to ∞ $$\sum_{n=1}^{\infty}\frac {n^2} {n!}$$
1
vote
4answers
110 views

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

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^...
5
votes
0answers
201 views

Finding sum to infinity: $\sum\limits_{n = 1}^{ \infty}\frac{n^2}{n!}$ [duplicate]

I am trying to find what this value will converge to $$\sum_{n = 1}^{ \infty}\frac{n^2}{n!}$$ I tried using the Taylor series for $e^x$ but couldn’t figure out how to manipulate it to get the above ...
0
votes
2answers
73 views

Sum of infinite squares and factorials: $\frac{1^2}{1!}+ \frac{2^2}{2!}+ \frac{3^2}{3!} + \frac{4^2}{4!} + \dots$ [duplicate]

$$\frac{1^2}{1!}+ \frac{2^2}{2!}+ \frac{3^2}{3!} + \frac{4^2}{4!} + \dotsb$$ I wrote it as: $$\lim_{n\to \infty}\sum_{r=1}^n \frac{(r^2)}{r!}.$$ Then I thought of sandwich theorem, it didn't work. ...
5
votes
3answers
119 views

Series representation of $2e$

According to GR9768, problem 37: $$\sum_{k=1}^{+\infty} \frac{k^2}{k!} = 2e$$ Can someone please explain how to get started in showing that?
7
votes
2answers
720 views

Why is this equation $\sum_{k=0}^{\infty}k^2\frac{\lambda^k}{k!e^\lambda}=\lambda +\lambda^2$ true?

Why is $$\sum_{k=0}^{\infty}k^2\frac{\lambda^k}{k!e^\lambda}=\lambda +\lambda^2$$ For the context: I am trying to calculate $E(X^2)$, where X is a poisson distributed random variable. All my ...
9
votes
1answer
180 views

How to prove that $\sum_{k=1}^{\infty}\frac{k^{n+1}}{k!}=eB_{n+1}=1+\cfrac{2^n+\cfrac{3^n+\cfrac{4^n+\cfrac{\vdots}{4}}{3}}{2}}{1}$

Through some calculation, it can be shown that $$e = 1+\cfrac{1+\cfrac{1+\cfrac{1+\cfrac{\vdots}{4}}{3}}{2}}{1}\tag{1}$$ $$2e = 1+\cfrac{2+\cfrac{3+\cfrac{4+\cfrac{\vdots}{4}}{3}}{2}}{1}\tag{2}$$ $$5e ...
3
votes
1answer
209 views

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 ...

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