# How to Test Whether a Series Converges or Diverges

Say you’re trying to figure out whether a series converges or diverges, but it doesn’t fit any of the tests you know. No worries. You find a benchmark series that you know converges or diverges and then compare your new series to the known benchmark.

If you’ve got a series that’s *smaller* than a *convergent* benchmark series, then your series must also converge. If the benchmark converges, your series converges; and if the benchmark diverges, your series diverges. And if your series is *larger* than a *divergent* benchmark series, then your series must also diverge. Here’s the mumbo jumbo.

**Direct comparison test:**

How about an example? Determine whether

converges or diverges. Piece o’ cake. This series resembles

which is a geometric series with *r* equal to

(Note that you can rewrite this in the standard geometric series form as

this series converges. And because

converge. Here’s another one: Does

converge or diverge? This series resembles

the harmonic *p*-series that is known to diverge. Because

must also diverge. By the way, if you’re wondering why this example considers only the terms where

here’s why:

**Feel free to ignore initial terms.** For any of the convergence/divergence tests, you can disregard *any* number of terms at the beginning of a series. And if you’re comparing two series, you can ignore any number of terms from the beginning of either or both of the series — and you can ignore a different number of terms in each of the two series.

This utter disregard of innocent beginning terms is allowed because the first, say, 10 or 1,000 or 1,000,000 terms of a series always sum to a finite number and thus never have any effect on whether the series converges or diverges. Note, however, that disregarding a number of terms *would* affect the total that a convergent series converges to.

(Are you wondering why this disregard of beginning terms doesn’t violate the direct comparison test’s requirement that

Everything’s copacetic because you can lop off any number of terms at the beginning of each series and let the counter, *n*, start at 1 anywhere in each series. Thus the “first” terms *a*_{1} and *b*_{1} can actually be located anywhere along each series. Make sense?)

**Fore!** (That was a joke.) The direct comparison test tells you *nothing* if the series you’re investigating is *greater* than a known *convergent* series or *less* than a known *divergent* series.

For example, say you want to determine whether

converges. This series resembles

which is a *p*-series with *p* equal to

The *p*-series test says that this series diverges, but that doesn’t help you because your series is *less* than this known divergent benchmark.

Instead, you should compare your series to the divergent harmonic series,

(it takes a little work to show this; give it a try). Because your series is *greater* than the *divergent* harmonic series, your series must also diverge.