The Line Is at p = 1, on the Divergent Side

You'll learnto prove that the harmonic series diverges in two lines, to run the same argument on a general exponent and see the threshold fall out of the antiderivative, to place the boundary case on the divergent side and keep it there, to read the exponent off a radical or a denominator rather than off the first number you see, and to use a p-series as the benchmark every comparison test in 10.6 will need.

Two topics have now leaned on the claim that the harmonic series diverges. Here it is proved, in two lines, and generalised: the sum of one over n to the p converges exactly when p is greater than 1. The inequality is strict, and the case sitting on the line is the harmonic series itself.

8 STEPS · 6 QUICK CHECKS · THE BOUNDARY IS THE WHOLE DIFFICULTY · v1

p greater than 1 converges · p = 1 is the harmonic series · strict

Before you start

The horizontal axis is read two ways

In steps 1 to 3 and 7 it is x, and the curves are functions of it. In steps 4 to 6 it is the exponent p, and the marks along it are whole series rather than points on a graph. Every one of those frames says which reading is in force, in the caption under the panel.

Comparing areas is safe here

A unit of y is drawn about 4.3 times taller than a unit of x, which stretches every area by the same factor — so the ratio of one shaded region to another is what it would be on square paper. Nothing in this file claims a slope, which is what that stretch would break.

Watch for

Step 2 asks you to tell two curves apart and you will not be able to. That is the point of the step rather than a failure of the drawing: one of the two areas is finite and the other is not, and nothing about how the curves look distinguishes them. The rule has to be known.

Step 1 / 8