Let . An antiderivative of is to be approximated by integrating a power…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Let r(x)=1+2x5. An antiderivative of r is to be approximated by integrating a power series.
(a) Obtain a power-series representation for r(x) about x=0 and write the first five non-zero terms.
(b) Integrate the series from part (a) term-by-term to obtain a power series for a particular antiderivative R(x) satisfying R(0)=0. Provide the first five non-zero terms of R(x). Write R(x) in sigma notation as well.
(c) Using the series for R(x), approximate R(41) with enough terms so that the absolute error of your approximation is less than 10−4. Show how you decide the number of terms required, and state both the approximation and the error bound.
(d) Justify that the error-bound method you used in part (c) is valid on the interval 0≤x≤41. Your justification should specifically mention the structure of the remainder for the series you integrated.