AP Calculus BC 10.10 Alternating Series Error Bound Questions

$f(-3)=\sum_{n=1}^{\infty} \frac{(n+1)(-3)^{n}}{n^{2} 6^{n}}=\sum_{n=1}^{\infty} \frac{(n+1)}{n^{2}}\left(-\frac{1}{2}\right)^{n}$ 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