If is positive, continuous, and decreasing for and , then and either both converge or both diverge.

To prove this theorem we first partition the interval into unit intervals. The total areas of the inscribed rectangles and the circumscribed triangles are as follows:

(inscribed area)

(circumscribed area)

The precise area under the graph of from to lies between the inscribed and circumscribed areas which implies . Using the nth partial sum, , we can write this inequality as . Now assuming that converges to L, it follows that for , . Consequently, is bounded and monotonic, and thereby by the Bounded Monotonic Sequence Theorem it converges. So, converges.

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