Is anything known about these integrals? Textbook suggestions are welcome
\begin{equation*} f(n,p)=\int_{x=-0.5}^p \frac{n!}{x!(n-x)!} dx, \end{equation*}
$n>0, p\le n+0.5$.
For instance, as $n$ grows $2^n-f(n,n+1/2)$ seems to be positive and monotonically decreasing...how would I find the limit?
ListPlot[Table[2^n - NIntegrate[n!/(x! (n - x)!), {x, -1/2, n + 1/2}], {n, Range[30]}], PlotRange -> All]