Statement of integral test

Suppose is positive and decreasing on . Then converges if and only if converges.

Proof

Let . By properties of the Riemann Integral, exists. Partition the interval by . We now have

As is positive and decreasing, it takes it's supremum at the left most point of each interval and it's infimum at it's right most point of each interval, for each . We now have

This is equivalent to the statement (by adjustment of index)

Now define for

and note that

, and we deduce that is monotone increasing. Now suppose that is finite. Then we have by the left hand side of the inequality derived above

and this means that is bounded above and monotone increasing, so the infinite sum converges. Now suppose that diverges. Then by the right hand side of the inequality we have

and this implies that diverges. This completes the proof.