Consider the rational function q(x)= 4-3x 2+5x

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the rational function q(x)=4−3x2+5x,q(x)=\frac{4-3x}{2+5x}, which is analytic at x=0x=0. (1) Rewrite q(x)q(x) in the form A+B 11+CxA + B\,\dfrac{1}{1+Cx}, where AA, BB, and CC are constants to be determined. (2) Using the form found in part (a), produce the Maclaurin series for q(x)q(x) and list all terms through and including x5x^{5}. Show all algebraic steps leading to the final expanded series. (3) Determine the radius of convergence of the series obtained in part (b) and justify your conclusion. (4) Using the series from part (b), evaluate lim⁡x→0q(x)−q(0)−q′(0)xx2.\lim_{x\to 0} \frac{q(x)-q(0)-q'(0)x}{x^{2}}. Show how the required limit follows directly from the coefficients of the series.

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