Question

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that
Rn(x) → 0.] Find the associated radius of convergence R.
f(x) = 8(1 − x)^−2
show step by step including finding the derivatives.

Answer 1

Recall that for |x| < 1, we have

$\displaystyle \frac1{1-x} = \sum_{n=0}^\infty x^n$

Differentiating both sides gives

$\displaystyle \frac1{(1-x)^2} = \sum_{n=0}^\infty nx^{n-1} = \sum_{n=0}^\infty (n+1)x^n$

and multiplying both sides by 8 gives the series for f(x) :

$f(x)=\displaystyle \frac8{(1-x)^2} = \boxed{8\sum_{n=0}^\infty (n+1)x^n}$

and this converges over the same interval, |x| < 1, so that the radius of convergence is 1.