I am looking for an approximation of $\sqrt{1 + x}$ or $\sqrt{x}$.
We know $\sqrt{1 + x}$ has a Taylor expansion:
${\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\cdots ,}$
But for the problem I am working on, (a problem in random walk), I would like to get an alternative approximation of $\sqrt{1 + x}$ or $\sqrt{x}$, and hope it has some natural connections to combinatorics or probability or random walk.
Is there an approximation of $\sqrt{1 + x}$ or $\sqrt{x}$ which has some connections to combinatorics or probability or random walk ?