The vertical displacement (in meters) of a lightly damped oscillator for small

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

Problem

The vertical displacement

y

(in meters) of a lightly damped oscillator for small, dimensionless time parameter

x

is modeled by the infinite series

y(x)=\sum_{n=0}^{\infty}(-1)^n\,\frac{\bigl(\tfrac{x}{2}\bigr)^{2n+1}}{2n+1}.

Answer the questions that follow. (a) Identify an elementary function that is equal to

y(x)

on its interval of convergence. (b) Determine the largest open interval, centered at

0

, for which the series representation converges. (c) Compute

y(1)

in simplest exact form. (d) Interpret the sign of

y(1)

in the context of the oscillator’s direction of displacement.