Hi !
Prove that :
$\displaystyle \lim_{n \to \infty} ~ e^{-n} \sum_{k=0}^n \frac{n^k}{k!}=\frac 12$
i think the limit originally comes from probability, something that i'm a complete idiot at it! there's the idea of the proof in this thread. i know a calculus proof but it's long and very ugly!
so i think it's a good challenge finding a neat calculus (analysis) solution for this question.
Yes, that was using a probability proof
Same type as the previous ones :P
Laurent in the thread gave the way to solve it, I'll detail it here
Consider a sequence $\displaystyle (X_n)_{n \geq 1}$ of independent random variables, following a Poisson distribution with parameter 1 :
$\displaystyle \mathbb{P}(X_i=k)=e^{-1} \cdot \frac{1}{k!}$
Then let's consider $\displaystyle S_n=X_1+\dots+X_n$
It is easy to prove that it follows a Poisson distribution with parameter n :
$\displaystyle \mathbb{P}(S_n=k)=e^{-n} \cdot \frac{n^k}{k!}$
We can see that :
$\displaystyle \mathbb{P}(S_n \leq n)=\sum_{k=0}^n e^{-n} \cdot \frac{n^k}{k!}$
But $\displaystyle \{S_n \leq n\}=\{S_n-n\leq 0\}=\left\{\frac{S_n-n}{\sqrt{n}} \leq 0\right\}$
Hence $\displaystyle \mathbb{P}\left(\frac{S_n-n}{\sqrt{n}} \leq 0\right)=e^{-n} \sum_{k=0}^n \frac{n^k}{k!}$
Then we must check the conditions for applying the Central Limit Theorem :
$\displaystyle \mathbb{V}\text{ar}(X_i)=\mathbb{E}(X_i)=1<\infty \Rightarrow X_i \in L^2$
And the $\displaystyle X_i$ are independent and identically distributed.
The Central Limit Theorem says that :
$\displaystyle \frac{S_n-n \mathbb{E}(X_i)}{\sqrt{n}}=\frac{S_n-n}{\sqrt{n}}$ converges to the standard normal distribution $\displaystyle \mathcal{N}(0,1)$
In particular, there is a convergence of the cumulative density functions :
$\displaystyle \mathbb{P}\left(\frac{S_n-n}{\sqrt{n}} \leq 0\right) \longrightarrow \int_{-\infty}^0 \frac{1}{\sqrt{2\pi}} e^{-t^2/2} ~dt$, which is $\displaystyle \frac 12$ because the integrand is even.
This finishes the proof :
$\displaystyle e^{-n} \sum_{k=0}^n \frac{n^k}{k!} \longrightarrow \frac 12$
Haaa... I really like these kinds of limits that use probability, it's just great
Now, if you have time, I'd be interested in seeing this proof you have that uses calculus ^^