# Using the Direct Comparison Test to Determine If a Series Converges

The direct comparison test is a simple, common-sense rule: If you’ve got a series that’s smaller than a convergent benchmark series, then your series must also converge. And if your series is larger than a divergent benchmark series, then your series must also diverge. Here’s the mumbo jumbo.

**D****irect ****C****omparison ****T****est:**** **

Piece o’ cake. This series resembles

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

Here’s another one: Does

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 1000 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.

The direct comparison test tells you *nothing* if the series you’re investigating is *bigger* than a known *convergent* series or *smaller* than a known *divergent* series.

For example, say you want to determine whether

The *p*-series test says that this series diverges, but that doesn’t help you because your series is *smaller* 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.