Problem 6.47. Prove the integral test as follows. Let $f$ be a continuous function such that $f(x) \ge 0$ and is decreasing for all $x \ge 1$, and let $f(n) = a_n$.
Suppose $\int_1^\infty f(x) dx$ converges.
1. Partition the interval $[1, N]$ into $N - 1$ equal intervals and sketch the rectangles as you did in Problem 6.43.
2. Express the total area of the rectangles in sigma notation; express the total area of the circumscribed rectangles in sigma notation.
3. Establish these two inequalities geometrically.
$\sum_{n=1}^N a_n \le a_1 + \int_1^N f(x) dx$ and $\int_1^N f(x) dx \le \sum_{n=1}^N a_n$
4. Prove that $\sum_{n=1}^\infty a_n$ converges by showing that $S_n$ is bounded above and increasing.
