§8.6 Quantifying the accuracy of approximations

Introduction

Alternating series of real numbers

Error Approximations for Taylor Polynomials

Summary

Consider the alternating geometric series

. We want to explore how the partial sums of the series compare to and approximate the exact sum of the series, which is

.

Recall that the $n$ th partial sum, $S_n$, is the sum of the first $n$ terms of the infinite geometric series $S$. This means that

, which we view as being in the form

. Note that the exact fractional values of $S_1$, $S_2$, $\ldots$, $S_6$, have been recorded below along with their decimal representations. Your task is to use this information to do some related computations that help us understand the behavior of the series and its partial sums.

First, by computing the differences between $S_n$ and $S$ for several different values of $n$ (recalling that $S = \frac{5}{9} = 0.\overline{5}$), fill in the first column of blank spaces provided below. In addition, fill in decimal representations of $a_3$, $\ldots$, $a_6$ to compare to the differences in the preceding adjacent column. For decimal representations you enter, be accurate to within $0.00001$.

Determine how well the $100^{\text{th}}$ partial sum $S_{100}$ of

approximates the value of the converging alternating series.

Use the fact that $\sin(x) = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \cdots$ to estimate $\sin(1)$ to within $0.0001$. Do so without entering “$\sin(1)$” on a computational device. After you find your estimate, enter “$\sin(1)$” on a computational device and compare the results.

Recall our recent work with $\int_0^1 e^{-x^2} \, dx$ which can be expressed as the series

. Use this series representation to estimate $\int_0^1 e^{-x^2} \, dx$ to within $0.0001$. Then, compare what a computational device reports when you use it to estimate the definite integral.

Find the Taylor series for $\cos(x^2)$ and then use the Taylor series and to estimate the value of $\int_0^1 \cos(x^2) \, dx$ to within $0.0001$. Compare your result to what a computational device reports when you use it to estimate the definite integral.

Recall we know that if $|x| \lt 1$, then

. What happens if $x = 1$?

Explain why the series $1 - \frac{1}{2} \cdot 1^2 + \frac{1}{3} \cdot 1^3 - \cdots + (-1)^{n-1} \frac{1}{n} \cdot 1^n + \cdots$ must converge and estimate its sum to within $0.01$. What is the exact sum of this series?

Determine the maximum error possible when using the degree $10$ Taylor polynomial centered at $a = 0$ for $\sin(x)$ to approximate the value of $\sin(2)$.

Use the degree $10$ Taylor polynomial (centered at $a = 0$) of $f(x) = e^x$ to estimate the value of $e^2$. What is the maximum error of your estimate, according to the Lagrange Error Bound? How does this compare to the actual error between $e^2$ and $T_{10}(2)$ as reported by a computer algebra system?

Use a degree $n$ Taylor polynomial (centered at $a = 0$) of $f(x) = \cos(x)$ to estimate the value of $\cos(1)$ within an accuracy of $0.00000001$. According to the Lagrange Error Bound, what value of $n$ is needed to achieve this accuracy? What is the resulting approximate value of $\cos(1)$?

Recall that for $f(x) = \ln(1+x)$, its Taylor series centered at $a = 0$ is given by

and that the $n^{\text{th}}$ derivative of $f(x) = \ln(1+x)$ is given by

. If we want to estimate $f(0.5) = \ln(1.5)$ to within an accuracy of $0.0001$, what value of $n$ is needed to achieve this accuracy from computing $T_n(0.5)$, according to the Lagrange Error Bound?

In this exercise we consider the definite integral

from two different perspectives.