The Integral Decides, and It Is Not the Sum

You'll learnto check the three hypotheses before running the test rather than after, to see why a falling curve is trapped between two sets of unit rectangles, to read the integral as a verdict on convergence and never as a value for the sum, and to reach for this test exactly when an antiderivative is available.

This is the first test with hypotheses that matter, and the first that can be run correctly and still be read wrongly. The integral answers one question, whether the series converges. It does not answer what the series sums to, and for the sum of one over n squared those two numbers are 1 and 1.6449. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · A VERDICT, NOT A VALUE · v2

Positive, falling, and integrable — then a verdict, never a value

Before you start

What you're looking at

A curve in pink, the terms it produces at the whole numbers, violet rectangles one unit wide standing on those steps, and the teal area under the curve itself.

The question

If you can measure the area, have you measured the sum, or only decided whether one exists?

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