<s>[INST] How to get the series expansion? I found a series expansion of $(1+2x)^{1/2x}$  is:
$$1+  \frac{1}{2x} 2x + \frac{\frac{1}{2x} (\frac{1}{2x}-1)}{2!} (2x)^2 +  \frac{\frac{1}{2x} (\frac{1}{2x}-1) (\frac{1}{2x}-2)}{3!} (2x)^3  $$
But which law and how they have got this expansion. Is it related to the the binomial theorem $(1+x)^n$? 
 [/INST] Your expression isn't true with the exponent $\frac1{2x}$ since in this case we should write
$$(1+2x)^{1/2x}=\exp\left(\frac1{2x}\log(1+2x)\right)$$
but if the expression has the form $(1+2x)^{1/2\alpha}$ with constant exponent then we have
$$(1+2x)^{1/2\alpha}=1+  \frac{1}{2\alpha} 2x + \frac{\frac{1}{2\alpha} (\frac{1}{2\alpha}-1)}{2!} (2x)^2 +  \frac{\frac{1}{2\alpha} (\frac{1}{2\alpha}-1) (\frac{1}{2\alpha}-2)}{3!} (2x)^3+\cdots $$ and it's just the Tayor expansion. 
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