Question

I'm interested in $\theta(N):=\int_0^1 (1-x)^{N-1} e^{xN} dx$. I'd like to show that $\theta(N)\sim c/\sqrt{N}$ as $N\to\infty$ and determine $c$. Any ideas?

Answer 1

Denote by $I$ your integral. Then, $I = e^N \int_0^1 {x^{N - 1} e^{ - xN} \,{\rm d}x} = \frac{{\Gamma (N)e^N }}{{N^N }}\int_0^N {\frac{{x^{N - 1} e^{ - x} }}{{\Gamma (N)}}\,{\rm d}x}.$ Now, if $X_1,\ldots,X_N$ are independent and identically distributed exponential(1) random variables, then their sum $X_1 + \cdots + X_N$ has gamma density $x^{N-1}e^{-x}/\Gamma(N)$, $x>0$. Thus, the last integral above is equal to ${\rm P}(X_1 + \cdots + X_N \leq N)$, or equivalently to ${\rm P}(X_1 + \cdots + X_N - N \leq 0)$. By the central limit theorem, ${\rm P}(X_1 + \cdots + X_N - N \leq 0) \to 1/2$ as $N \to \infty$ (since the $X_i$ have expectation equal to $1$). So, using $N^N \sim N!e^N /\sqrt {2\pi N} $, we get $ I \sim \sqrt {\frac{\pi }{2}} \frac{1}{{\sqrt N }}.$

Answer 2

Or just type Series[Integrate[(1 - x)^N Exp[x N], {x, 0, 1}], {N, Infinity, 1}] into Mathematica, and preserve those valuable neurons.