p-series are infinite series of the form

p series model
P series are important functions in math and science, so it's worth getting to know them. When p = 1 the p-series is called the harmonic series. Notice that we start this kind of series with n = 1 in the summation because we don't want a zero denominator.

This section relies on a fair bit of calculus. We'll start by looking at the p-integral.

The p-integral

The p-integral is the integral of f(x) = x-p, were p is a constant. For p = 1 we know that the integral is ln|x| + C, but in order to examine the convergence or divergence of p-series.

The integral is a simple one (I've flipped the exponent and denominator of the integral from -p+1 to 1-p for convenience).

p integral equation 1

Now of course this is an improper integral, so we have to evaluate it using the limit as our "dummy" variable R approaches .

p integral equation 2

The second term above is finite because a is a constant. The first is trickier. Depending on the value of p, it will either converge to a limit or diverge to infinity.

p integral equation 3

Here's the full limit for the case where p > 1:

p integral equation 4

and for p < 1:

p integral equation 5

We already said that when p = 1 the integral is ln|x| + C, which approaches as R→∞. With the p-integral in hand, we can use the integral test to determine which p-series converge.

The harmonic series

An important p-series with p = 1

The harmonic series is

harmonic series equation 1

To determine whether the harmonic series converges, we let f(x) = 1/x and integrate the function from 1 to ∞. Clearly the series is trapped under f(x) as drawn, so if the function converges to a finite limit, so must the series. On the other hand, if the function diverges, then the series won't converge to a fixed limit either.

harmonic series graph

This integral is the special case of the p-integral with p = 1. It goes like this:

harmonic series equation 2

Because the integral that traps the series below it does not reach a finite limit (diverges), the series diverges.

Comparison of the harmonic series with a convergent p-series.

It might be a little difficult to come to terms with the notion that the harmonic series doesn't eventually sum to a limit even thought the limit of the nth term is zero, but it's true.

The denominator of the terms of the harmonic series just doesn't get large "fast enough" for the series to converge.

On the right is a comparison of the sums

harmonic series comp. eq. 1

The left column of each colored block is just the nth term. The right column is the cumulative sum. When a series converges, it is clear that successive terms modify digit positions more and more to the right of the decimal. That's not true of the harmonic series.

The graph on the right shows how each of these sums grow. The convergent p-series approaches a horizontal asymptote near y = 1.6. While the growth of the harmonic series slows down as x grows, the series is unbounded, and will continue to grow without converging to a finite limit.

harmonic series comparison table

harmonic series comparison graph


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