Binomial Theorem

$1+\frac{1}{2 x}-\frac{1}{2!}\left(\frac{1}{2 x}\right)^2+\frac{1 \cdot 3}{3!}\left(\frac{1}{2 x}\right)^3-\frac{1 \cdot 3 \cdot 5}{4!}\left(\frac{1}{2 x}\right)^4+\ldots \infty$

$\frac{1}{\sqrt{x}}+\frac{1}{2} \sqrt{x}-\frac{1}{2!} \frac{x \sqrt{x}}{2^2}+\frac{1 \cdot 3}{3!} \frac{x^2 \sqrt{x}}{2^3}-\ldots . \infty$

$1+\frac{1}{\sqrt{x}}+\frac{1}{2} x \sqrt{x}+\frac{1}{2!} \frac{x^2 \sqrt{x}}{2^3}+\frac{1 \cdot 3}{3!} \frac{x^3 \sqrt{x}}{2^4}+\ldots . \infty$

$\frac{1}{\sqrt{x}}+\frac{1}{2 x \sqrt{x}}-\frac{1}{2!}\left(\frac{1}{2 x}\right)^2 \frac{1}{\sqrt{x}}+\frac{1 \cdot 3}{3!}\left(\frac{1}{2 x}\right)^3 \frac{1}{\sqrt{x}}-\ldots \ldots \infty$