Problem

g, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 1, plus, x, start superscript, 2, end superscript, end fraction
We know that
g, left parenthesis, x, right parenthesis, equals, 1, minus, x, start superscript, 2, end superscript, plus, x, start superscript, 4, end superscript, minus, x, start superscript, 6, end superscript, plus, point, point, point
for x(1,1)x\in(-1,1).
Using this fact, find the power series for f, left parenthesis, x, right parenthesis, equals, minus, start fraction, 2, x, divided by, left parenthesis, 1, plus, x, start superscript, 2, end superscript, right parenthesis, start superscript, 2, end superscript, end fraction.
Please choose from one of the following options.