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 decreasing 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, to recognise the two ways a function fails the hypotheses and what "eventually" buys you, 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.

8 STEPS · 6 QUICK CHECKS · A VERDICT, NOT A VALUE · v1

positive · continuous · decreasing — then integrate, then conclude

Before you start

What you're looking at

A function plane, not the index panel the last three topics used — the horizontal axis is x and takes every value, because this test is about a continuous function laid against the terms it produces. Pink is the function, violet is a rectangle family, and teal is the area under the curve.

Comparing areas is safe here

A unit of y is drawn about 3.8 times taller than a unit of x, which stretches every area by the same factor — so the ratio of the rectangles' area to the curve's is exactly what it would be on square paper. Nothing in this file claims a slope or a tangent, which is what that stretch would break.

Watch for

In step 4 the integral evaluates to exactly 1 and the series sums to 1.6449. Watch the teal bar and the violet bar in the card: the test is correct, the arithmetic is correct, and the two numbers are different. That gap is what the test trades away for being computable.

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