AP Calculus BC 10.10 Series Error Bounds Overview

Review series error bounds by applying the first omitted term rule to control alternating-series approximation error and report a guaranteed bound.

[Maximum number: 2]

The Maclaurin series for a function f is given by $\sum_{n=1}^{\infty} \frac{(n+1) x^{n}}{n^{2} 6^{n}}$ and converges to f(x) for all x in the interval of convergence. It can be shown that the Maclaurin series for f has a radius of convergence of 6.

It can be shown that $f(-3)=\sum_{n=1}^{\infty} \frac{(n+1)(-3)^{n}}{n^{2} 6^{n}}=\sum_{n=1}^{\infty} \frac{n+1}{n^{2}}\left(-\frac{1}{2}\right)^{n}$ and that the first three terms of this series sum to $S_{3}=-\frac{125}{144}$ . Show that $\left|f(-3)-S_{3}\right|<\frac{1}{50}$ .

$f(-3)=\sum_{n=1}^{\infty} \frac{(n+1)(-3)^{n}}{n^{2} 6^{n}}=\sum_{n=1}^{\infty} \frac{(n+1)}{n^{2}}\left(-\frac{1}{2}\right)^{n}$ is an alternating series with terms that decrease in magnitude to 0.

By the alternating series error bound, $\sum_{n=1}^{3} \frac{n+1}{n^{2}}\left(-\frac{1}{2}\right)^{n}=-\frac{125}{144}$ approximates f(-3) with error of at most
$\left|\frac{4+1}{4^{2}}\left(-\frac{1}{2}\right)^{4}\right|=\frac{5}{256}<\frac{5}{250}=\frac{1}{50}.$ Thus, $\left|f(-3)-S_{3}\right|<\frac{1}{50}$ .

Uses fourth term

1 point

Scoring notes:

- The first point is earned for correctly using x=-3 in the fourth term. (Listing the fourth term as

part of a polynomial is not sufficient.) Using x=-3 in any term of degree five or higher does not

earn this point.

- The expression $\frac{4+1}{4^{2}}\left(-\frac{1}{2}\right)^{4}$ earns the first point, but just $\frac{5}{256}$ does not earn the first point.

- A response including the expression $\frac{4+1}{4^{2}}\left(-\frac{1}{2}\right)^{4}$ that is subsequently simplified incorrectly earns

the first point but not the second.

- To earn the second point the response must state that the series for f(-3) is alternating or that the

alternating series error bound is being used.

○ A response of just "Error $\leq \frac{4+1}{4^{2}}\left(-\frac{1}{2}\right)^{4}<\frac{1}{50}$ " (or any equivalent mathematical expression)

earns both points, provided it is accompanied by an indication that the series is alternating.

- A response that declares the error is equal to $\frac{5}{256}$ (or any equivalent form of this value) does not

earn the second point.

Total for part (b) 2 points

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Covered syllabus points

10.10 Alternating Series Error Bound question 1