AP Calculus BC Topic 10.6: Comparison Tests for Convergence

Topic 10.6 key terms

harmonic series: The series $\sum_{n=1}^{\infty}\frac1n = 1+\frac12+\frac13+\cdots$ . Its terms go to 0, but it diverges: its partial sums grow without bound (bundle the terms into groups that each add at least 1/2).
alternating harmonic series: The series $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} = 1-\frac12+\frac13-\cdots$ . It converges (to ln 2), even though the harmonic series diverges.
p-series: A series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ . It converges if p > 1 and diverges if p ≤ 1.
direct comparison test: If $0\le a_n\le b_n$ for all n (from some point on): if $\sum b_n$ converges, so does $\sum a_n$ ; if $\sum a_n$ diverges, so does $\sum b_n$ . Below a divergent series or above a convergent one proves nothing.
limit comparison test: If $a_n>0$ , $b_n>0$ and $\lim_{n\to\infty}\frac{a_n}{b_n}=L$ with $0<L<\infty$ , then $\sum a_n$ and $\sum b_n$ both converge or both diverge.
benchmark series: A series whose behavior you already know (a p-series or a geometric series) that you compare with. For a rational term, keep the biggest power of n on top and on the bottom to choose it.

Transcript

1,293 words · 4 parts

Lift the stage

Real stages are raised, so let’s lift this one three meters above the phones. Every light path gets longer, so every fan in the line sends less light than before, and the nearest fans lose the most.

Fan n stands n meters out, and the stage is three meters up, so Pythagoras makes its distance squared n squared plus nine. It sends one over n squared plus nine. We don’t know the new total exactly, and we don’t need to, because every fan got dimmer, so the new total has to stay under the old ceiling.

That’s the direct comparison test. If every term of your series is positive and sits below the matching term of a series that converges, then your series converges too. It also works the other way around. If your series sits above the terms of a series that diverges, then yours diverges too, just like the harmonic ribbon sitting on top of all those halves. In a picture, our smaller partial sums never overtake the bigger ones, so they can never get past the bigger one’s ceiling.

Let’s write the raised line the way you’d write it on the exam. For every n from one on, one over n squared plus nine is positive and less than one over n squared. The sum of one over n squared is a p-series with p equal to two, which is bigger than one, so it converges. That means our series converges by the direct comparison test.

Now let’s do one together, a series that showed up on the 2024 AP exam, the sum of n plus one over n squared. It’s tempting to see the n squared on the bottom and call it a p-series with p equal to two. But the n on top matters. Split the fraction into one over n plus one over n squared, and now you can see it’s bigger than one over n. Every term is bigger than one over n, and the harmonic series diverges, so this series diverges by direct comparison.

There’s one trap here that catches a lot of students. Being smaller than a divergent series tells you nothing. One over two n squared and one over two n both sit under one over n, and one of them converges while the other diverges. Their partial sums both start out under the harmonic sums. Slide N out and find out which one breaks through the ceiling of two.

Being bigger than a convergent series tells you nothing either, because two over n and two over n squared both sit above one over n squared. For a ceiling you need something bigger that converges, and for a floor you need something smaller that diverges.

Far away, it’s the same crowd

Now raise the stage over the rings. Ring n still holds n times the fans of ring one, but each of them is farther away. Ring n now sends n over n squared plus nine, which is less than one over n, and every ring is dimmer than before. But that’s the useless direction we just talked about, smaller than a series that runs off to infinity, so direct comparison can’t tell us whether the raised rings still blow up.

Let’s compare the rings one at a time instead. Raised over flat, ring n keeps n squared over n squared plus nine of its light. The first ring keeps a tenth, because the stage is right above it, and the third ring keeps half. But ring ten keeps about ninety-two percent, and ring one hundred keeps more than ninety-nine point nine percent, because a fan a hundred meters back hardly notices three meters of stage.

Try it yourself. Drag the probe out across the field and find a ring that keeps at least ninety-nine percent of its light. Far out, the raised rings are basically the same as the flat rings, and the flat rings add up to infinity, so the raised rings have to add up to infinity too.

That’s the limit comparison test. If both series have positive terms, and the limit of a n over b n is a finite, positive number, then the two series do the same thing. They both converge or they both diverge.

Why does it work? Far out, the ratio stays between half of L and two L, so your terms sit between half L and two L times the benchmark’s. A constant times the benchmark does whatever the benchmark does, and your series is squeezed between two of those.

On paper, take a n to be n over n squared plus nine, and b n to be one over n, and both are positive. Their ratio is n squared over n squared plus nine, and as n goes to infinity that goes to one.

One is finite and positive, and the harmonic series diverges, so the raised rings diverge by the limit comparison test. Don’t read anything into the number one itself. L only has to be finite and positive, and then your series copies whatever the benchmark does, so here it diverges because the harmonic series does.

The hard part is choosing the benchmark, and the trick is to keep only the biggest power of n on top and on the bottom. For three n plus two over n cubed plus five, that leaves three n over n cubed, three over n squared, and the three just ends up inside L, so the benchmark is one over n squared.

The ratio with one over n squared goes to three, which is finite and positive, so that series converges right along with the sum of one over n squared. Your turn to choose. Which benchmark would you compare with this series?

Full credit

Let’s do one exactly the way the exam asks it, with a justification. The biggest powers are two n squared on top and n to the fourth on the bottom, so it acts like two over n squared, and our benchmark is one over n squared. A full-credit answer has four pieces. First, both series have positive terms, and two n squared minus one is positive for every n from one on.

Second, write the limit of a n over b n with real limit notation and evaluate it. Multiplying by n squared gives two n to the fourth minus n squared over n to the fourth plus three n, and that goes to two. Third, write down the words the grader is looking for, that two is finite and positive.

Fourth, finish with the conclusion and the name of the test. The sum of one over n squared converges because it’s a p-series with p equal to two, so our series converges by the limit comparison test. Notice that the two at the end didn’t mean anything special. It only had to be finite and positive.

Now you’re the grader. Which of these justifications earns full credit for the sum of one over n squared plus five n?

The promise, kept

The harmonic series diverges, its alternating cousin converges, a p-series converges exactly when p is bigger than one, and the two comparison tests borrow the answer from a series you already know. Let’s go back to the festival, because now you can prove which crowd is which. The rings make the harmonic series, so an endless field of fans would pour infinite light onto this stage.

The single line makes a p-series with p equal to two, so even infinitely many fans add up to less than twice the first one. Lifting the stage doesn’t change either answer, by direct comparison for the line and by limit comparison for the rings. If you want a puzzle for tonight, look up Olbers’ paradox, which is our crowd in three dimensions.

Topic 10.6: Comparison Tests for Convergence