Limit of Series: Proving lim n→∞ ∑e^-n n^k/k! = 1/2

[Apasz]

Homework Statement

prove that lim n→∞ of $\sum$$^{n}_{k=0}$ e$^{-n}$ n$^{k}$ / k! = 1/2

The Attempt at a Solution

I seem to be mishandling the series. After taking n→∞, the sum of (n^k)/k! is just the taylor series expansion of e^n. Then I should get e^(-n)*e^n = 1.

Where am I going wrong??

 

[micromass]

You're messing up your variables.

You say "After taking the limit $n\rightarrow +\infty$". But you seem to interpret that only has "making the sum infinite".

The expressions $e^{-n}$ and $n^k$ also depend on n. So if you take the limit as $n\rightarrow +\infty$, then those things don't stay the same.

Allow me to totally butcher mathematics for a moment, but it's to make things clear. If you take the limit $n\rightarrow +\infty$ of your expression then you don't end up with

$$e^{-n}\sum_{k=0}^{+\infty}\frac{n^k}{k!}$$

Rather, you would end up with (please forgive me)

$$e^{-\infty}\sum_{k=0}^{+\infty}\frac{\infty^k}{k!}$$

The above of course makes no sense. But I think it makes the situation clear.

 

[STEMucator]

Prove that what? That the series converges?

 

[Apasz]

Sorry, prove the limit = 1/2 (could have sworn I had written it).

Micromass, thanks. I would be lying if I said I didn't suspect that was the problem, it still kind of makes me uneasy.

Anyways now I am trying to replace the sum by an integral (Euleur-Maclaurin), and I get:

$\sum$$^{n}_{k=0}$ e$^{-n}$$\frac{n^k}{k!}$ = 1/2 + $\frac{e^{-n}}{2}$$\frac{n^n}{(n!)}$ + $\int$$^{n}_{0}$ e$^{-n}$$\frac{n^k}{k!}$dk

so I have hopes because the 1/2 is there to stay, the 2nd term probably goes to zero after taking the limit, same with the integral except I don't know how to handle the factorial in the integral. Now I am reading on Gamma functions.. maybe that will help.

 

[micromass]

It may sound like a strange question, but do you know the Central Limit Theorem and Poisson distributions?

 

[Apasz]

It may sound like a strange question, but do you know the Central Limit Theorem and Poisson distributions?

No, I haven't learned them.. although googling Poission distributions.. it seems it is related to my problem.

In applying for a Masters in Fluid Dynamics (Math) after getting a Physics Undergrad, one of my potential supervisors gave me a set of problems to 'check me out'. I killed most of them without too much sweat and tears, but this one really I'm having a hard time with.

Anyways,I will read about Poisson distributions and Central Limit Theorem.