Obtain the first three non-zero terms of the expansion of $(1+2x)^{-1}$ $$\frac{x}{\sqrt{1+2x}}$$

Question

Obtain the first three non-zero terms of the expansion of $(1+2x)^{-1}$ $$\frac{x}{\sqrt{1+2x}}$$ in ascending powers of $x$ descending powers of $x$. In each case, find the range of $x$ for which the expansion is valid.

Accepted Answer

(a)(i) $1-2x+4x^2, |x| \frac{1}{2}$; (b)(i) $x-x^2+\frac{3}{2}x^3, |x| \frac{1}{2}$.

Binomial series expansion

Binomial series expansion