MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The function f is defined by f(x)=1−5x1.
(a) Rewrite f(x) as an infinite series of powers of x that converges for x near 0. Clearly indicate the first four non-zero terms using sigma notation or expanded form.
(b) Provide a justification, based on the structure of your series in part (a), for the radius of convergence of the series. State the corresponding interval of x-values on which the series representation is valid.
(c) Explain, without additional calculation, how the coefficients of xn in your series would change if f were replaced by g(x)=1+5x1. Illustrate your explanation with the first three non-zero terms of the new series.