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) = 6(1 − x)−2

Answer 1

I guess the function is

$f(x)=\dfrac6{(1-x)^2}$

Rather than computing derivatives of $f$, recall that for $|x|<1$, we have

$g(x)=\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n$

Notice that

$g'(x)=\dfrac1{(1-x)^2}$

so that $f(x)=6g'(x)$. Then

$f(x)=6\displaystyle\sum_{n=0}^\infty nx^{n-1}=6\sum_{n=1}^\infty nx^{n-1}=6\sum_{n=0}^\infty(n+1)x^n$

also valid only for $|x|<1$, so that the radius of convergence is 1.