So-called *p-*series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A *p-*series is of the form

(where *p* is a *positive* power). The *p*-series for *p* = 1 is called the *harmonic* series. Here it is:

Although this grows *very* slowly — after 10,000 terms, the sum is only about 9.79! — the harmonic series in fact diverges to infinity.

By the way, this is called a *harmonic* series because the numbers in the series have something to do with the way a musical string like a guitar string vibrates — don’t ask. For history buffs, in the 6th century b.c., Pythagoras investigated the harmonic series and its connection to the musical notes of the lyre.

Here’s the convergence/divergence rule for *p*-series:

As you can see from this rule, the harmonic series forms the convergence/divergence borderline for *p*-series. Any *p*-series with terms *larger* than the terms of the harmonic series *diverges*, and any *p*-series with terms *smaller* than the terms of the harmonic series *converges*.

The *p*-series for *p* = 2 is another common one:

The *p*-series rule tells you that this series converges. It can be shown that the sum converges to

But, unlike with the geometric series rule, the *p*-series rule only tells you whether or not a series converges, not what number it converges to.