AP Calculus BC 10.4: Integral Test

$e^{-x}$ is positive, decreasing, and continuous on the interval $[0, \infty)$.

Conditions

To use the integral test to show that $\sum_{n=0}^{\infty} \frac{1}{e^{n}}$ converges, show that
$\int_{0}^{\infty} e^{-x} d x$ is finite (converges).

Improper integral

$\int_{0}^{\infty} e^{-x} d x=\lim _{b \rightarrow \infty} \int_{0}^{b} e^{-x} d x=\lim _{b \rightarrow \infty}\left(-\left.e^{-x}\right|_{0} ^{b}\right)=\lim _{b \rightarrow \infty}\left(-e^{-b}+e^{0}\right)=1$
Because the integral $\int_{0}^{\infty} e^{-x} d x$ converges, the series $\sum_{n=0}^{\infty} \frac{1}{e^{n}}$
converges.

Evaluation

Scoring notes:

- To earn the first point a response must list all three conditions: $e^{-x}$ is positive, decreasing, and continuous.

- The second point is earned for correctly writing the improper integral or for presenting a correct limit equivalent to the improper integral (for example, $\lim _{b \rightarrow \infty} \int_{0}^{b} e^{-x} d x$ ).

- To earn the third point a response must correctly use limit notation to evaluate the improper integral, find an evaluation of $e^{0}$ (or 1 ), and conclude that the integral converges or that the series converges.

- If an incorrect lower limit of 1 is used in the improper integral, then the second point is not earned. In this case, if the correct limit (1 / e) is presented, then the response is eligible for the third point.

- If the response only relies on using a geometric series approach, then no points are earned [0-0-0].

- A response that presents an evaluation with $\infty$, such as $e^{-\infty}=0$, does not earn the third point.

Total for part (a) 3 points