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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

Instructor:
*Michael Gundlach*

Do you remember working with polynomials? Imagine a polynomial that is infinitely long. Instead of having a finite degree, it has an infinite degree. Such an infinite polynomial is called a power series. In this lesson, we examine traits of power series.

In the past, you've probably worked with polynomials, or functions that look like this:

Imagine that instead of there being some highest power on *x*, (like 5 in this function), there was no largest power. For example, consider the following function:

The ... at the end tells us that this polynomial goes on forever, continuing in the same pattern. Such infinite polynomial functions are called **power series**.

In order to avoid using the ... notation, which sometimes makes it less clear what the coefficients on each power of *x* are, we usually represent these infinite polynomials using summation notation:

The Î£ at the beginning tells us we will be summing up terms according to some pattern described to the right of the Î£. The *n* = 0 below the Î£ tells us we will start by putting 0 in for *n* in the formula to the right of Î£. If there is a finite number on top of Î£, we stop adding up terms when we get to that number. Since we have infinity on top of Î£, we know that the function sums up an infinite number of terms, and thus is a power series.

All of these equations are some more concrete examples of the use of this notation:

Occasionally we run into power series of the following form:

Here are some examples:

Take a moment and see if you can figure out what *a* is in each of the power series. Have you figured it out? In the first, it's 2, the second, 1, and the third, -3 (since -(-3) = 3). In such a power series, *a* is sometimes called the center of the power series. Sometimes power series with non-zero centers are called **power series about a**.

At this point, you might have been wondering, 'If we're adding up an infinite number of terms, how do we know that the sum isn't infinite?' Mathematicians have come up with some ways to check that, but first we need to discuss terminology. If an infinite sum adds up to a finite number, we say it **converges**. When it comes to power series, since power series are functions (i.e. there is a variable, *x*, in which we can plug numbers to get an output), we want to know all the numbers which we can put into *x* that give us a convergent sum.

It turns out that the numbers for which a power series converge always form an interval, which we (unsurprisingly) call the **interval of convergence**. It also turns out that the endpoints of this interval will always have *a*, the center of the power series, as its midpoint. Thus, we call the absolute value of the difference between *a* and either endpoint of the interval of convergence the **radius of convergence**.

In this lesson, we're only going to talk about how to find the radius of convergence, since there are some nuances in finding the full interval of convergence that are beyond the scope of this lesson. However, we're also going to use the radius of convergence to find an open interval contained in the interval of convergence.

The main tool mathematicians use to determine the radius of convergence is the **ratio test**. If we have a power series

then the ratio test says that the radius of convergence, *r*, can be found by evaluating the limit:

Using the radius *r*, we can determine that the open interval (*a - r, a + r*) is contained in the interval of convergence.

Here's an example. Suppose *h(x)* is a function such that

In this function,

Thus,

As you can see, once we simplify everything down, we get the answer of *r* = 5. Thus, we say that *h(x)* has a radius of convergence of 5. Thus, its interval of convergence contains the open interval (-1,9).

Occasionally, the ratio test tells us that the radius of convergence is 0. In cases like this, the interval of convergence is just the center, since if we put the center of the series in for *x*, all but the first term goes to 0, meaning the series converges (since you're adding up a single number plus a bunch of zeros). For example, consider this following series of:

When we do the ratio test, we get the following:

Thus, the interval of convergence for this series is 6, the set {6}.

On the other hand, if the radius of convergence is infinite, we say the interval of convergence is the whole real line, or (-âˆž,âˆž). An example of such a power series is the following:

When we do the ratio test, we end up taking the limit as *n* goes to infinity of *n*, which is âˆž. Thus, the radius of convergence of this power series is âˆž, and it had an interval of convergence of (-âˆž,âˆž)

Let's take a few moments to review what we've learned about functions defined by power series. In this lesson, we learned that a **power series** is essentially an infinite polynomial. It has a **center**, *a*, and **converges**, meaning an infinite sum adds up to a finite number. It converges on its **interval of convergence**, which is when the number for which a power series converge and always form an internal.

The interval of convergence contains the open interval (*a - r, a + r*), where *r* is the **radius of convergence**, or the absolute value of the difference between *a* and either endpoint of the interval of convergence. The main tool mathematicians use to determine the radius of convergence is the **ratio test**, which will help us check our work.

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

- What is L'Hopital's Rule? 7:11
- Applying L'Hopital's Rule in Simple Cases 7:53
- Applying L'Hopital's Rule in Complex Cases 8:13
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- How to Calculate a Geometric Series 9:15
- Power Series in X & the Interval of Convergence 5:49
- Functions Defined by Power Series 6:17
- Go to L'Hopital's Rule, Integrals & Series in Calculus

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