A Maclaurin polynomial is used to approximate the value of with the third–degree…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A Maclaurin polynomial is used to approximate the value of e0.3 with the third–degree polynomial P3(x)=1+x+2!x2+3!x3.
(a) Identify two distinct theoretical techniques from the study of power‐series remainders that could, in principle, provide an upper bound for ∣e0.3−P3(0.3)∣. Do not compute any numerical values at this stage.
(b) Explain which of the two techniques named in part (a) is not applicable to this approximation and state clearly why its hypotheses fail for the given series.
(c) For the technique that remains applicable, describe qualitatively (no calculation required) how increasing the degree of the Taylor polynomial from 3 to 5 would affect the numerical size of the bound on the remainder.