AP Calculus BC 10.6: Comparison Tests

Use the limit comparison test with the series $\sum_{n=0}^{\infty} \frac{1}{e^{n}}$ to show that the series $g(1)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 e^{n}+3}$ converges absolutely.

$\lim _{n \rightarrow \infty}$$\frac{(-1)^{n}}{2 e^{n}+3} e^{n}=2$
Sets up limit 1 point The limit exists and is positive. Therefore, because the series $\sum_{n=0}^{\infty} \frac{1}{e^{n}}$ converges, the series $\sum_{n=0}^{\infty}\left|\frac{(-1)^{n}}{2 e^{n}+3}\right|$ converges by the limit comparison test. Thus, the series $g(1)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 e^{n}+3}$ converges absolutely.

Scoring notes:

- The first point is earned for setting up the limit comparison, with or without absolute values. Limit notation is required to earn this point.

- The reciprocal of the given ratio is an acceptable alternative; the limit in this case is 1/2.

- The second point cannot be earned without the use of absolute value symbols, which can occur explicitly or implicitly (e.g., a response might set up the limit comparison initially as

- Earning the second point requires correctly evaluating the limit and noting that the limit is a positive number. For example, L=2>0 or L=1 / 2>0. Therefore, comparing the limit L to 1 does not earn the explanation point.

- A response does not have to repeat that $\sum_{n=0}^{\infty} \frac{1}{e^{n}}$ converges.

- A response that draws a conclusion based only on the sequence (such as $\frac{1}{e^{n}}$ ) without referencing a series does not earn the second point.

- If the response does not explicitly use the limit comparison test, then no points are earned in this part.

- A response cannot earn the second point for just concluding that "the series" converges absolutely because there are multiple series in this part of the problem. The response must specify that the series g(1) or $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 e^{n}+3}$ converges absolutely.

Total for part (b) 2 points