Asymptotic approximation of binomial theorem

Question

Binomial theorem is a very popular theorem that: $$(x + y) ^ n = \sum_{i=0}^n {n \choose i}x^i y^{n-i}$$

I am looking for any papers (the newer the better) where I can find any informations about asymptotic approximation of the binomial theorem, especially for the case when I have $(1 + y)^n$. Here I found the approximation, that the asymptotic approximation of $(1+x)^k$ is $1 + kx + {k \choose 2}x^2 + {k \choose 3}x^3 + O(x^4)$. But maybe are any works where I can find more accurate approximation?

Answer 1

The asymptotic approximation you have is just binomial expansion up to the 4th term. What happens as $n \to \infty$ depends on $y$. If $y$ is a constant larger than $0$, the sum diverges, if $-1<y<0$ it converges to 0. If $y= \frac{a}{n}$, the limit is of course $e^a$. Hence, no single answer exists.

Answer 2

It was mentioned in the comments that the author is seeking an approximation in the case that $n \to \infty$ and $y \approx 1/n$. One approach is to set $y = \frac{1+z}{n}$, so that the expression is

$$ \left(1 + \frac{1+z}{n}\right)^n. $$

Then we can expand the function

$$ f(a,b) = (1 + (1+a)b)^{1/b} $$

in Taylor series about the point $(a,b) = (0,0)$, obtaining

$$ f(a,b) = e \left(1 + a - \frac{b}{2} + \frac{a^2}{2} - \frac{3ab}{2} + \frac{11 b^2}{24} + O(3)\right), $$

where $O(3)$ represents terms of cubic order. Taking $a = ny-1$ and $b = 1/n$ we arrive at the approximation

$$ \left(1 + y\right)^n \approx e \left(\frac{1+n^2y^2}{2}+\frac{1}{n}-\frac{3 y}{2}+\frac{11}{24 n^2}\right). $$

By the same method, an approximation including the third order terms is

$$ \left(1 + y\right)^n \approx e \left( \frac{2+3ny+n^3y^3}{6}+\frac{5}{12 n}-\frac{7 n y^2}{12}-\frac{y}{3}-\frac{7}{36 n^2}+\frac{47 y}{72 n}-\frac{7}{16 n^3} \right). $$