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Functions Defined by Power Series

Functions Defined by Power Series
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  • 0:04 Series and Power Series
  • 0:44 Notation
  • 1:30 Center of a Power Series
  • 1:59 Convergence
  • 5:30 Lesson Summary
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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.

Series and Power Series

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


A generic polynomial


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:


An infinite polynomial


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.

Notation

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:


Generic power series


The Σ at the beginning tells us we will be summing up terms according to some pattern described to the right of the Σ. The n = 0 below the Σ tells us we will start by putting 0 in for n in the formula to the right of Σ. If there is a finite number on top of Σ, we stop adding up terms when we get to that number. Since we have infinity on top of Σ, 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:


Power Series Examples


Center of a Power Series

Occasionally we run into power series of the following form:


Power series centered at a


Here are some examples:


Off center power series


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.

Convergence

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

Power series centered at a


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