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.

Recall that for |x| < 1, we have

Differentiating both sides gives

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

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

d

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~coco

sine

step-by-step explanation: