If is decreasing, then and either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.
For a nonnegative nonincreasing function , comparison with unit-width rectangles givesOn , the function is at most ; on , it is at least . Summing these inequalities gives the bounds. Applying them to gives for the harmonic numbers. Unlike a convergence verdict alone, the finite bounds estimate the size of a partial sum.
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The Integral Test for convergence is a method used to determine whether a series converges or diverges by comparing it to an improper integral. It applies specifically to series that consist of positive, decreasing functions. ### Statement of the Integral Test Let \( f(x) \) be a positive, continuous, and decreasing function for \( x \geq N \) (where \( N \) is some positive integer).