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Integrals & derivatives of functions with known power series

AP.CALC: LIM‑8 (EU), LIM‑8.D (LO), LIM‑8.D.6 (EK), LIM‑8.G (LO), LIM‑8.G.1 (EK)

Problem

g, left parenthesis, x, right parenthesis, equals, start fraction, 4, divided by, 1, minus, 2, x, end fraction
We know that
g, left parenthesis, x, right parenthesis, equals, 4, plus, 8, x, plus, 16, x, squared, plus, 32, x, cubed, plus, point, point, point
for x, \in, left parenthesis, minus, start fraction, 1, divided by, 2, end fraction, comma, start fraction, 1, divided by, 2, end fraction, right parenthesis.
Using this fact, find the power series for f, left parenthesis, x, right parenthesis, equals, start fraction, 8, divided by, left parenthesis, 1, minus, 2, x, right parenthesis, squared, end fraction.
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