How to Test Whether a Series Converges or Diverges

By Mark Ryan

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:

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How about an example? Determine whether

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

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which is a geometric series with r equal to

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

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this series converges. And because

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converge. Here’s another one: Does

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converge or diverge? This series resembles

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

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

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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

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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 a1 and b1 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

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converges. This series resembles

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which is a p-series with p equal to

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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,

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(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.