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 watch the threshold fall out of the antiderivative, to keep the boundary case on the divergent side, and to read the exponent off a radical or a denominator rather than off the first number you see.

Two topics have now leaned on the claim that the harmonic series diverges. Here it is proved, in two lines, and then generalized: 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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE BOUNDARY IS THE WHOLE DIFFICULTY · v2

Above p = 1 it converges; on the line and below it does not

Before you start

What you're looking at

A curve, the area under it in teal, and a row of dots below the axis on some beats where the horizontal axis is read as the exponent rather than as a position.

The question

Two curves can look the same and enclose different amounts of room; how would you ever tell?

Step 1 / 12