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. ] f(x) = 5(1 − x)−2

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 n x^(n-1) = \sum_(n=0)^\infty (n+1) x^n`

so that the Maclaurin expansion of the given function is

`\displaystyle \frac5{1-x} = \boxed{\sum_(n=0)^\infty 5(n+1) x^n} = 5 + 10x + 15x^2 + 20x^3 + \cdots`