Topic 10.13: Radius and Interval of Convergence of Power Series

Every term makes it worse

This might be the calmest curve in all of math. It’s one over one plus x squared, a smooth hill that rises to one and settles gently back toward zero on both sides, and nowhere along the whole road does it do anything strange.

You can build this exact hill out of nothing but powers of x, with a polynomial that never ends. Start with one, subtract x squared, add x to the fourth, subtract x to the sixth, and keep going forever. At one half, every term we add pulls the thread closer to the hill, and the sum settles right onto zero point eight, exactly the height of the curve.

But at two, where the curve is a calm zero point two, the sums go one, negative three, thirteen, negative fifty-one, two hundred five, and every new term makes it worse. With twenty terms it hugs the hill almost all the way from negative one to one, and outside that it tears away in both directions, as if it hit a wall.

But there’s no wall at one. The curve doesn’t do anything there at all, so what’s stopping this series? By the end of this lesson, you’ll be able to look at any power series and say where it works and where it falls apart, right down to its two edges. You’ll also see the invisible thing that’s breaking this one. Let’s start with what this thing actually is.

A polynomial that never ends

The board tells the same story, with the curve in blue and our first few terms in orange, close at one half and nowhere near each other at two. We call a sum like this a power series. It’s the sum, from n equals zero to infinity, of a sub n times x minus c, to the n. We call the a sub n’s the coefficients, and we build the whole series around the center, c.

We center ours at zero, with terms negative one to the n times x to the two n, so every odd power has coefficient zero. The moment you pick a value for x, it stops moving, and only n marches off to infinity. That means at any one x, a power series is just an ordinary series of numbers, the kind you already know how to test.

Out on the road, a probe at zero point three freezes x right there, and with twenty terms the sum hops up and down and settles onto the curve. Now you drive. Try a few spots on the road, run the terms, and pin one spot where the sum settles and one where it grows, as close to the edge as you can.

Let’s prove where that edge is. Freeze x at one half and the series becomes one, minus a quarter, plus a sixteenth, and so on, where each term is the one before it times negative one quarter.

That’s a geometric series with ratio negative one quarter, so it adds up to one over one minus r. That’s one over one plus a quarter, which is four fifths, or zero point eight, exactly the height of the curve. At two the ratio is negative four, bigger than one in size, so the terms grow and the sum never settles.

For any x the ratio is negative x squared, so the series converges exactly when x squared is less than one, which means x is between negative one and one, the edge you found. The distance from the center to the edge is the radius of convergence. Here it’s one. All the x’s where the series converges form the interval of convergence. The center is always in it, since every term after the first is zero there.

At x equals one the curve’s height is one half. What do you think the series does there?

The ratio test finds the radius

Most power series aren’t geometric, so we need the ratio test. Let’s try it on the sum of x minus two to the n, over n times three to the n, which we build around the center two. Call each whole term u sub n. The ratio test takes the limit, as n goes to infinity, of the absolute value of u sub n plus one over u sub n.

If that limit is less than one the series converges, if it’s more than one it diverges, and exactly one tells you nothing. We write the next term times the flip of this one, and almost everything cancels, so one x minus two and one three are left over, giving the absolute value of x minus two over three, times n over n plus one.

As n runs to infinity, n over n plus one goes to one, while x minus two doesn’t move, because x is a number we picked. So the limit is the absolute value of x minus two, over three.

That’s less than one when x is within three of the center, between negative one and five, so the radius of convergence is three. The center, two, sits right in the middle of negative one and five, which tells us we solved it right. But right at negative one and right at five, the limit is exactly one, the one case where the ratio test goes silent.

Only the center, or everywhere

Out on the road, that first series reached from negative one to one. Every power series has a reach, but it isn’t always a stretch of road like this one. The sum of x to the n over n factorial reaches all the way to the horizon, so it converges for every x. Put the factorial upstairs, n factorial times x to the n, and the reach collapses to one point, the center, where every power series has to converge.

The ratio test shows why. With the factorial downstairs the ratio is the absolute value of x over n plus one, which goes to zero for every x, so the radius is infinite. With it upstairs the ratio is n plus one times the absolute value of x, which blows up unless x is zero, so the radius is zero.

Every other power series lands in between, with a finite radius and two edges to check by hand, and those are the only three possibilities. Where does the sum of n factorial times x minus three to the n converge?

The two edges

Back on the road with our ratio test example, its reach runs from negative one to five, and the two white rings mark the edges the ratio test couldn’t decide. Both edges are three units from the center. Which edges do you think let the series in?

At five, every term becomes one over n, the harmonic series, and the sums climb forever. After a million terms they’ve barely passed fourteen, but they never stop, so that edge is out. At negative one, the terms keep their sizes, but the signs alternate. The sums zigzag in and settle near negative zero point six nine, right on the curve this series adds up to. That edge is in.

On paper you start from the open interval the ratio test gave us, and then you test each edge by putting it into the original series. At five the three to the n’s cancel and you get the harmonic series, which diverges. At negative one you get negative one to the n over n, and since one over n decreases to zero, the alternating series test says it converges.

So the interval of convergence runs from negative one, included, up to five, not included. We write that with a square bracket at negative one and a round one at five. This is the justification a grader wants to see. The ratio limit is the absolute value of x minus two over three. That’s less than one exactly when x is between negative one and five, so the series converges there.

At negative one it’s the alternating harmonic series, which converges by the alternating series test, because one over n decreases to zero. At five it’s the harmonic series, which diverges. So the interval of convergence is from negative one, included, to five, not included. Series free-response questions often ask for this kind of argument, and the edges carry their own points.

Gravity’s series

This next power series has real units in it. You know this rocket, and as it climbs, gravity gets weaker. Call the Earth’s radius R sub E. At a height of h kilometers, gravity is nine point eight one meters per second squared, times R sub E over R sub E plus h, squared. As a power series around the launch pad, it’s nine point eight one times one, minus two h over R sub E, plus three times h over R sub E squared, and so on, so the coefficients count up with alternating signs.

Let’s find its radius together. The next term over this one is n plus two over n plus one, times the absolute value of h over the Earth’s radius. What do you think that goes to as n grows? The n plus two over n plus one goes to one, so the limit is the absolute value of h over the Earth’s radius. That’s less than one when h stays within six thousand three hundred seventy-one kilometers of the pad.

At both edges the terms grow like n plus one, so both edges are out. And that radius of convergence, six thousand three hundred seventy-one kilometers, is the radius of the Earth itself. GPS satellites orbit about twenty thousand two hundred kilometers up. What does this series do at that height?

The formula divides by zero one Earth radius straight down from the pad, at the center of the Earth. A power series reaches the same distance both ways, so it also stops one Earth radius up, at this shell. The space station, about four hundred twenty kilometers up, is well inside, where four terms pin gravity at eight point six three meters per second squared, and the GPS satellites are far outside, where the sums explode.

Your turn

Now try one on your own. It’s the sum of two x minus one, to the n, over n squared. Run the ratio test and find the radius of convergence. Now test the two edges. What’s the interval of convergence?

Calculus on a series

Power series have one more surprise. When its radius is positive, a power series is the Taylor series of the function it adds up to on its open interval. That means each coefficient is the nth derivative at the center, over n factorial. So our geometric series is the Taylor series of one over one plus x squared, and its coefficients know every derivative at zero.

Differentiate the series twice and the x squared term becomes two factorial times its coefficient, while every later term still has an x in it, so at zero only that one survives. Our x squared coefficient is negative one, so the second derivative at zero is negative two, and we never had to differentiate the fraction. Use that to find the fourth derivative of one over one plus x squared at zero, without taking a single derivative.

You can also integrate term by term. The integral of our curve from zero to x is arctangent of x, and integrating one, minus t squared, plus t to the fourth, one term at a time, gives x, minus x cubed over three, plus x to the fifth over five, and so on. The radius is still one. Integrating or differentiating a power series never changes its radius.

The edges can change, though. At x equals one, the series is one, minus a third, plus a fifth, minus a seventh, and those sizes decrease to zero, so it converges by the alternating series test.

Negative one gives the same series with every sign flipped, so this time both edges are in. Arctangent of one is pi over four, so watch four times the sum creep toward three point one four one five nine. It’s pi. Madhava found that formula for pi in India around fourteen hundred, Leibniz found it again almost three centuries later, and it falls right out of our calm curve.

So integrating pulled both edges in. Differentiating can push one out, so let’s take our ratio test example, which worked from negative one up to five, and differentiate it term by term. Each power n comes down and cancels the n underneath. What’s the interval of convergence of the derivative?

What’s hiding

So we’ve proved this series stops at one. But we still haven’t found anything on the road at one that could stop it. Usually a fraction breaks where its denominator is zero, but one plus x squared is never zero for a real x, because x squared can’t be negative one. But the road is only one line through a much bigger world.

The imaginary numbers you met in algebra, like i, live off to the side of it. If x steps sideways into them, one plus x squared hits zero at i and at negative i. The height here shows the size of the function’s value, and near i and negative i that size shoots up to infinity, two enormous peaks, each one unit from the center.

The ratio test only ever measured the distance from x to the center. That means a power series only knows how far you are from its center, not which way you went. It reaches equally far in every direction, a circle, until it touches the nearest peak, and that’s why we call it a radius. Let the series build its own landscape and the circle appears by itself, a close match inside and a wall outside. Our road crosses it from negative one to one, the very interval we proved.

Now you move the center. Drag it along the road until the series can reach all the way to two. That’s where it fell apart at the start. Now see how little you can get away with. Slide the center back toward zero and find the smallest center that still reaches two.

To find where a power series works, you usually start with the ratio test for the radius. Then you test each edge by hand, in the original series. The radius can be zero, a number or infinity, and differentiating or integrating keeps it but can change the edges.

For the curves we met today, the radius was the distance to the nearest trouble. For gravity, that was the center of the Earth. For our calm curve, it was two peaks hiding just off the road. You came in watching a calm curve break an infinite polynomial. Now you can say where a power series works, right down to its two edges.