The top equation is very clear, but how is the first approximation done? The author mentions this line as an alternative to Stirling's approximation. It should be a straightforward equation, but sorry I could not get it
Thank you in advance!
$$ \begin{split} 1 &= \sum_K \binom{N}{K}2^{-N} \\ &\approx 2^{-N} \binom{N}{N/2} \sum_{r=-N/2}^{N/2} e^{-(r/\sigma)^2/2}\\ &\approx 2^{-N} \binom{N}{N/2} \sigma \sqrt{2\pi}. \end{split} $$
The last approximation comes because for large $N$, $$ \sum_{r=-N/2}^{N/2} e^{-(r/\sigma)^2/2} \approx \int_\mathbb{R} e^{-(r/\sigma)^2/2} dr = \sigma \int_\mathbb{R} e^{-u^2/2} du = \sigma\sqrt{2\pi} $$ where $u = r/\sigma$ and the last step is a famous Gaussian integral, which can be taken, e.g., by switching to polar coordinates...