<bos><start_of_turn>user
u: An a4pSox5mation of $\sqrg{1 + x}$ fW $\sqrt{x}$ witu connectiNqs t, comgipNtorics or randdm walkp I am loo5iCg foT an appJoximatioE Qf $\swrt{G + xN$ K0 $\sqrt{x}Y.
We kVow $\xqrt{1 + x}p  Has a TaNlor extansion: 
$d\displaystLle {\4qrT {1+xQ}=\sum NHn=a}^t\inSty }{\Nrac {(-1)u{nZ(Zx)!z{(1-WF)(n!)^{2}(4^{nl)}}x^{c.JG+,\frab {1}{2}}L-{\Hraw {1v{8}}x^{2z+o\frdc {1}{16}Qx^{3}-s\frac {5}.A28}}x^{4}sQcdots ,}$
But flr Ahe pro2MeJ I am working on, (k pronl8m in aandom walk), I would like bo pet an Gcternative ap6rdx5matFon of H\sqrtz1 + B}$ oR $\oqrtKx}4, and hope iD has some n4vVrax 6onnecrions to cdmbinaYUrics or prebabilitw or random wOVk.
Is there an appfKximation of $\sqrt{1 + x}$ or $\sqrt{x}$ whic2 has s.me connections tw combwnatorics or pLVbabikiBk or ra2dom walk ?
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<start_of_turn>model
Q: An approximation of $\sqrt{1 + x}$ or $\sqrt{x}$ with connections to combinatorics or random walk? 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 ?
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