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Absolute and Conditional Convergence

Calculus 2 · Chapter 7: Convergence Tests for SeriesSection 7.4 · lesson 35 of 5410 min · free

← The Alternating Series TestNext lesson: The Ratio Test →

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In this lesson

Two ways to converge, and why the difference matters.

  • Strip the signs and test the absolute-value series
  • Use absolute convergence when sign patterns are irregular
  • See why rearrangement is safe for one kind and not the other
  • Classify with a disciplined two-part check

Transcript

Everything said in the video, section by section (1,424 words). Read along, or search it.

MAT-265 - Calculus 2 · Section 7.4

Absolute and Conditional Convergence

Section seven point three ended with a distinction that deserves its own treatment. A convergent series can owe its convergence to two very different things. Either its terms shrink fast enough on their own. Or they don't — and the series only converges because of sign cancellation. Telling those apart matters more than it might sound. Because one kind behaves like a finite sum, reliable and order-independent. And the other is so fragile that reordering its terms can change the answer to anything you like.

Two Kinds

Strip the Signs and Look Again

The way to tell them apart is simple. Strip the signs, and look again. Form the series of absolute values, and ask whether that converges. Let's classify two series. First. The sum of minus one to the n minus one, over n squared. Form the series of absolute values by stripping the signs. You get the sum of one over n squared. That's a p-series with p equal to two. Which converges. Since the absolute-value series converges, the original series is absolutely convergent. Its convergence does not rely on the alternating signs at all. Take the signs away and it still converges. Second. The alternating harmonic series. The series itself converges — we showed that in the last section, by the Alternating Series Test. Now strip the signs. You get the sum of one over n. The harmonic series. Which diverges. So the original converges, but its absolute-value series does not. That series is conditionally convergent. It owes its convergence entirely to sign cancellation. So the definitions are: absolutely convergent when the absolute-value series converges. Conditionally convergent when the series converges but its absolute-value series doesn't.

The Central Theorem

Absolute Convergence Implies Convergence

Now the central theorem of the section. And it makes absolute convergence the stronger, more desirable property. If the sum of absolute a n converges, then the sum of a n converges. Equivalently: every absolutely convergent series is convergent. The converse is false. A convergent series need not be absolutely convergent — the alternating harmonic series shows that. Let's see why the theorem holds, because the proof is short and clever. It rests on one squeezing inequality. For any real number a n, the value a n plus absolute a n sits between zero and two absolute a n. Why? Because when a n is positive, a n plus absolute a n equals two absolute a n. And when a n is negative, they cancel and it equals zero. So in all cases it's trapped between zero and two absolute a n. Now suppose the sum of absolute a n converges. Then the sum of two absolute a n converges too — a constant multiple of a convergent series, by the linearity of section six point five. By the Comparison Test, the nonnegative series sum of a n plus absolute a n converges, since its terms are bounded above by that convergent series. Finally, write the original series as a difference of two convergent series. Sum of a n equals sum of a n plus absolute a n, minus sum of absolute a n. Both pieces converge. So by linearity their difference converges. Which completes the argument.

The Practical Advantage

It Works for Any Sign Pattern

The practical advantage of this theorem is considerable. Checking whether the absolute-value series converges is a legitimate way to establish convergence of the original — even for series with irregular or unpredictable sign patterns. And testing the absolute-value series brings the entire positive-term toolkit to bear. The Comparison Test. The Ratio Test. The Integral Test. The p-series result. So rather than inventing special methods for each sign pattern, you convert the problem into a positive-term question that the existing tests already handle. Here's the case that makes the point. Determine whether the sum of sine n over n squared converges. The signs of sine n follow no simple alternating pattern at all. Sine of one, sine of two, sine of three — the signs jump around unpredictably. So the Alternating Series Test does not apply. It's not an alternating series. Instead, test for absolute convergence. Since the absolute value of sine n is at most one for every n, the absolute value of the term is at most one over n squared. And the sum of one over n squared is a convergent p-series. So by the Comparison Test, the absolute-value series converges. Hence the original series converges absolutely. And by the Key Result, it converges. Absolute convergence is the tool of choice whenever sign patterns are too irregular for the Alternating Series Test.

Why It Matters

Rearrangement

So why do we draw this distinction at all? Why not just say a series converges and leave it there? Because the two types behave differently under one of the most basic operations imaginable. Reordering the terms. For an ordinary finite sum, order never matters. Two plus five plus three equals three plus two plus five. You might expect the same of infinite sums. But that intuition holds only for absolutely convergent series. An absolutely convergent series can be rearranged in any order whatsoever, and it will still converge to the same sum. It behaves exactly like a finite sum in this respect. Reliable. Order-independent. A conditionally convergent series enjoys no such protection. The Riemann rearrangement theorem states that the terms of a conditionally convergent series can be reordered to make the series sum to any real number. Or even to diverge. The same collection of terms, in different orders, can total zero. Or one hundred. Or minus seven. That astonishing fragility is the practical reason the distinction is drawn. Absolute convergence certifies that a series is safe to manipulate. Conditional convergence is a warning that ordinary algebraic intuition can fail. Here's the practical version. Suppose a computation requires summing the alternating series with one over n cubed, and you want to add its terms in a convenient non-sequential order to speed up the calculation. Is that safe? Check absolute convergence. The magnitudes give a p-series with p equal to three. Which converges. So the series is absolutely convergent, and rearranging does not change the sum. The reordering is safe. Had it been only conditionally convergent, no such reordering could be trusted.

Classification

A Two-Part Check

The classification comes down to a table with four rows. If the series converges, and the absolute-value series converges: absolutely convergent. If the series converges, but the absolute-value series diverges: conditionally convergent. If neither converges: divergent. And the fourth row is the impossible one. The absolute-value series converging while the original diverges cannot occur — because absolute convergence forces convergence. That's the theorem. Let's classify one fully. The sum of minus one to the n, over root n. First check the series itself with the Alternating Series Test. b n is one over root n, which decreases to zero. So the series converges. Now check absolute convergence. Strip the signs and you get the sum of one over n to the one half. That's a p-series with p equal to one half. Which is at most one. So it diverges. The series converges, but not absolutely. So it is conditionally convergent.

Key Takeaways

The Pitfall, and What to Carry Forward

The most common error here is treating the implication as reversible. Absolute convergence implies convergence. But convergence does not imply absolute convergence. The alternating harmonic series is the standing counterexample. A student who concludes the series converges, so it converges absolutely, has reversed the theorem. A second error is declaring a series divergent because its absolute-value series diverges. But the divergence of the absolute-value series only rules out absolute convergence. Conditional convergence is still possible. So the original series must be tested separately — often with the Alternating Series Test. The correct procedure is a two-part check. Test the absolute-value series first. If it converges, the series is absolutely convergent and you're done. If it diverges, test the original series on its own — to distinguish conditional convergence from outright divergence. Skipping that second test misclassifies many conditionally convergent series as divergent. Five takeaways. A series is absolutely convergent if the sum of absolute a n converges. And conditionally convergent if it converges but the sum of absolute a n diverges. Absolute convergence implies convergence — the central theorem — but the converse is false. Absolute convergence is the stronger property. It works for any sign pattern, making it the tool of choice when the Alternating Series Test doesn't apply. Absolutely convergent series can be rearranged freely. Conditionally convergent series cannot — reordering can change the sum to anything. And classify with the two-part check. Never skip the second test. Next up is section seven point five — the Ratio Test. Which measures a series against a geometric benchmark.

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FAQ

Questions students ask us

About Absolute and Conditional Convergence

What is the difference between absolute and conditional convergence?
A series \sum a_n is absolutely convergent if \sum |a_n| converges, and conditionally convergent if \sum a_n converges but \sum |a_n| diverges. For example, \sum (-1)^{n-1}/n^2 is absolutely convergent because \sum 1/n^2 is a p-series with p=2. The alternating harmonic series converges by the Alternating Series Test, but \sum 1/n diverges, so it is conditionally convergent.
How do you test convergence of a series with irregular signs like sin n over n squared?
Test the series for absolute convergence, since absolute convergence implies convergence for any sign pattern. For \sum \sin n/n^2, the signs jump around unpredictably, so the Alternating Series Test does not apply. But |\sin n| \le 1, so |\sin n/n^2| \le 1/n^2, and \sum 1/n^2 converges. By the Comparison Test the absolute-value series converges, so the original series converges absolutely.
Can you rearrange the terms of a convergent infinite series without changing its sum?
Only if the series is absolutely convergent. By the Riemann rearrangement theorem, the terms of a conditionally convergent series can be reordered to sum to any real number or even to diverge. Absolutely convergent series behave like finite sums, so any rearrangement converges to the same sum. Before reordering terms, as in \sum (-1)^n/n^3, check absolute convergence. Here \sum 1/n^3 converges, so rearranging is safe.
If the absolute value series diverges, does the original series diverge?
No, a divergent absolute-value series only rules out absolute convergence, and the original series may still converge conditionally. For \sum (-1)^n/\sqrt{n}, the series \sum 1/n^{1/2} is a p-series with p=1/2\le 1, so it diverges. But b_n = 1/\sqrt{n} decreases to zero, so the Alternating Series Test shows the original converges. Always test the original series separately once the absolute-value series diverges.

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