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)=\frac6{(1-x)^2}`

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

`g(x)=\frac1{1-x}=\displaystyle\sum_(n=0)^\infty x^n`

Notice that

`g'(x)=\frac1{(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.