I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out:
$$P_N(s) = \begin{cases} \frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} &\text{N $\land$ s even or N $\land$ s uneven }\\0&\text{else}\end{cases} $$
for $N,s \in \mathbb{N}, \ s \le N$. The task is to show that $P_N(s) = \frac{1}{\sqrt{2\pi N}} \text{e}^{-\frac{1}{2} \frac{s^2}{N}}$ for big N.
Now I have thought about a Taylor-Approximation but I miserably fail when facing the expanded form of $P_N(s)$:
$$\binom{N}{\frac{1}{2}(N + s)} = \frac{1}{2^N} \frac{N!}{\left[\frac{1}{2}(N + s)\right]!\left[N - \frac{1}{2}(N + s)\right]!} = \frac{1}{2^N} \frac{N!}{\left[\frac{1}{2}(N + s)\right]!{\left[\frac{1}{2}(N - s)\right]!}}$$
So my problem is that I don't see how to differentiate that - and it remains a mistery for me how I am supposed to get the e-function in there....
When I look at the for big N, I don't see any simplifications either.
I am sorry for not being able to present any useful own thoughts - I really have been thinking about this forever....
Thank you very much for your help,
FunkyPeanut
You can approximate the binomial by a Gaussian:
Binomial[n, s]/2^n ~ Exp[-(n/2 - s)^2 /2/(n/4)]/Sqrt[2 Pi n/4]
$ \frac{\binom{n}{\frac{n}{2}+\frac{s}{2}}}{2^n}$ to $\frac{\exp \left(-\frac{\left(\frac{n}{2}-\left(\frac{n}{2}+\frac{s}{2}\right)\right)^2}{\frac{2 n}{4}}\right)}{\sqrt{\frac{2 \pi n}{4}}}$
and the right side simplifies to $\frac{\sqrt{\frac{2}{\pi }} e^{-\frac{s^2}{2 n}}}{\sqrt{n}}$
which differs by a factor of 2 from your P_n(s) above.