Topic 10.3: The nth Term Test for Divergence

Two towers

Here’s a chalk cliff high above the water, and right on its edge sits a stack of four identical books, with no glue and nothing hidden, and the top book hangs all the way out over the sea. Every time you slide one more book in underneath and push the whole stack out as far as it’ll go without tipping, the top reaches a little farther out over the water.

Look at what each new book adds, though. Every push is smaller than the one before, and the pushes keep shrinking toward nothing, which makes you think the reach has to level off somewhere. Right beside it stands a twin, and its builder follows a more careful rule. Its pushes shrink toward nothing too, and its first two books sit exactly where the first tower’s do.

Yet this one will never get past a single book length, not with a thousand books and not with a billion. That leaves a real question about the first tower. Is there an invisible wall out there for it too, or can it keep reaching as far as you want? If you took precalc with us, you’ve stood on this cliff before, and you watched this tower lean farther and farther out, but we never actually proved that it never stops.

By the end of this lesson you’ll be able to prove which of these towers keeps going, and you’ll work out how many books you need to be sure it hangs a full meter out over the sea. Let’s start with a single book and count exactly what each new one adds.

What one book adds

Every book is the same, twenty-four centimeters long, and we’ll treat them as perfectly balanced. With one book, you can slide it out until its middle sits right on the edge, which puts the tip half a book length out. Slide a second book in underneath, and the whole stack can move out another quarter of a length before it tips.

The third book adds a sixth, the fourth adds an eighth, and if you keep going, the nth book adds one over two n of a book length. There are two different numbers here, and this whole lesson depends on keeping them apart. One is the push from a single book, and we’ll call it a sub n. The other is the total reach, all those pushes added up, and we’ll call it S sub n, the partial sum.

With four books the total is twenty-five twenty-fourths, a little more than one whole book length, and that’s why the top book clears the edge. The twin’s builder is more careful, and pushes each new book out only half as far as the one before. Its pushes go a half, a quarter, an eighth, a sixteenth, so the first two match the tower exactly. Both lists of pushes shrink toward zero, but the twin’s shrink faster, because every one is half the last.

Here’s our tower on a graph, with each push as a bar on the left and the running total as a staircase on the right. For our tower the pushes shrink toward zero while the total keeps climbing, and after a thousand books it’s still climbing, at about three point seven book lengths.

You can already answer this one from the last lesson. If the twin’s builder keeps halving forever, what’s the farthest its tip can ever reach, in book lengths?

The twin’s graph tells a different story. Its pushes shrink toward zero too, but its total flattens out and hugs one, the wall it can never pass. That means the pushes going to zero can’t be what decides it, because both towers do that, and they end up in completely different places.

The nth-term test

The pushes can tell you one thing for sure, and you find it by turning the question around. Suppose a total does settle down at some number S, the way the twin settles at one. Every push is just the difference between two totals in a row, the total after n books minus the total after n minus one books. If the totals settle at S, both of those totals get closer and closer to S, so their difference gets closer and closer to S minus S, which is zero.

In tower language, if a tip ever settles at one spot, the pushes have to fade away to nothing, because a push that didn’t fade would keep shoving the tip out. That means if the pushes don’t fade, the tip can’t settle, and that’s the nth-term test. If the terms don’t go to zero, or their limit doesn’t exist, the series diverges.

The trap almost everyone falls into is the backward version, which says that when the terms do go to zero, the series converges. That’s false. When the limit of the terms is zero, the nth-term test is inconclusive, which means it tells you nothing at all. Our leaning tower is the counterexample, with pushes that go to zero and a reach that never settles, and in a few minutes we’ll prove it.

Let’s use it on a real series, the sum of three n plus two over five n minus one. To see where the terms go, divide the top and the bottom by n. The two over n and the one over n both shrink to zero, so the terms head to three fifths.

Every term ends up close to point six, and adding about point six over and over, forever, can’t possibly settle. On the exam you’d write it as one sentence. Since the limit of the terms is three fifths, which isn’t zero, the series diverges by the nth-term test.

Let’s do one together, the sum of negative one to the n. The terms go negative one, one, negative one, one, so they never settle on any number, and their limit doesn’t exist. These terms never settle down at zero, because they keep switching between negative one and one, so this series diverges too, even though its partial sums just bounce between negative one and zero. Diverging doesn’t have to mean shooting off to infinity.

Try the nth-term test on these three series. Which one has terms that let you prove the series diverges? A student sees that one over n squared goes to zero, and uses the nth-term test to claim that the series converges. Does that argument work?