Let be the function given by

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

Problem

Let

q

be the function given by

q(x)=\dfrac{1}{(1-3x)^{2}}

, defined for all real

x

satisfying

|x|<\dfrac{1}{3}

. (a) Derive a general expression for the

n

th power of

x

in the Maclaurin series of

q

. Express the coefficient in closed form using

n

and factorial notation. (b) Using your result from part (a), obtain an exact formula for

q^{(n)}(0)

valid for every non-negative integer

n

. Show the algebraic connection between the series coefficient and the derivative value. (c) Determine the radius of convergence of the Maclaurin series of

q

and justify your conclusion rigorously. (d) Let

T_{3}(x)

denote the third–degree Maclaurin polynomial of

q

. For

0\le x\le 0.2

, establish an upper bound for the absolute error

|q(x)-T_{3}(x)|

by means of an appropriate remainder estimate. Present your bound as a numerical inequality valid on the stated interval.

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