Convergence and Divergence Question 2

For $x\geq 1$ , the continuous function $g$ is decreasing and positive. A portion of the graph of $g$ is shown above. For $n\geq1$ , the $n$ th term of the series $\displaystyle \sum_{n=1}^\infty a_n$ is defined by $a_n=g(n)$ . If $\displaystyle \int\limits_1^\infty g(x) dx$ converges to 8, which of the following could be true?

\displaystyle \sum_{n=1}^\infty a_n=6

\displaystyle \sum_{n=1}^\infty a_n=8

\displaystyle \sum_{n=1}^\infty a_n=10

\displaystyle \sum_{n=1}^\infty a_n

diverges

Approach

By the integral test, both the improper integral and series converges.

The value of the series is:

\sum_{n=1}^\infty a_n = g(1) + g(2) +g(3)+...+g(n)

This is equivalent to a left riemann sum. The area under the curve is given as 8, so the sum of the series must be greater than 8. Only the third choice satisfies this.