How to find the Maclaurin series of $ \frac{7x}{1+2x}$

Question

How can you find the Maclaurin series of ${7x\over 1+2x}$

Answer 1

Hint: $\frac{1}{x+1}=1-x+x^2-x^3+ \cdots$.

Answer 2

$f(x) = \frac{7x}{1+2x} = \frac{3.5(1+2x) - 3.5}{1+2x} = 3.5 -\frac{3.5}{1+2x}$

Note that $\frac{1}{1+x} = 1-x+x^2-...$ with $|x|<1$

Then $f(x) = 3.5 - 3.5\cdot[1-(2x)+(2x)^2-(2x)^3+...] = 3.5[-(2x)+(2x)^2-(2x)^3+...]$

$$f(x) =3.5\sum_{r=1}^{\infty}(-2x)^r, |x|<\frac{1}{2}$$