Taylor's series expansion $\sqrt{\frac{1+x}{1-2x}}$

Question

$$\sqrt{\frac{1+x}{1-2x}}$$ expand as a series in ascending powers of x including the term in x^2 using first three non-zero terms to estimate $$\sqrt{\frac{3}{2}}$$

How do I solve this question?

Would my x be $\frac{1+x}{1-2x}^{\frac{1}{2}}$?

Answer 1

It is an easy matter to obtain

$$\sqrt{1+x}\approx1+\frac x2-\frac{x^2}8$$ and

$$\frac1{\sqrt{1-2x}}\approx1+x+\frac{3x^2}2.$$

Then the product is (keeping the terms up to quadratic)

$$1+\frac{3x}2+\frac{15x^2}8$$ and you can evaluate at $x=\dfrac18$.