## Calculus 8th Edition

Published by Cengage

# Chapter 11 - Infinite Sequences and Series - 11.11 Application of Taylor Polynomials - 11.11 Exercises - Page 821: 25

Four terms

#### Work Step by Step

We are given that $f(x)=e^x$ and $f^n(x)=e^x$ Check the Taylor inequality at $x=0.1$ Since, we have $|R_n(x)|\leq \dfrac{M}{(n+1)!}|x-a|^{n+1}$ This gives: $|R_2(x)|\leq \dfrac{e^{(0.1)}}{(2+1)!}|0.1-0|^{(2+1)}\approx 0.0002$ and $|R_3(x)|\leq \dfrac{e^{(0.1)}}{(3+1)!}|0.1-0|^{(3+1)}\approx 0.000005$ It has been seen that the remainder $|R_3(x)| \lt 0.00001$ require four terms.

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