Taylor Series

You know that formula; we've all seen it. You know, the one long, dreaded formula that teachers have been trying to grind into our heads for years (unsuccessfully, I may add), assuring us that it will be the cornerstone of our math-related careers. It's in every calculus, differential equation, and god-only-knows-what-else math book we could think of. It's the Taylor Series!

Have you ever actually wondered what it means? Well, if you have, then this is just the tutorial for you. For years, students have been mindlessly plugging numbers into that formula praying they come out with the right answer. Well, that's just fine and dandy until later on down the line, in whatever career path you choose, it creeps back up on you. You don't think it will happen, but believe me, it will! I don't know about you, but I can never remember formulas without getting to the core of what they actually do. So let's try and answer the question: just what does this darn formula do anyway?

Okay, so let's say that we have some obscure-looking function that we don't know much about. Imagine we only know the equation of the function, and it looks pretty ugly. If we could possibly approximate this ugly function by using polynomials, then for many purposes, the simple polynomial approximation might be good enough. Well, that's the information the Taylor Series can give you!

Let's call our point and let's define a new variable that simply measures how far we are from ; call the variable .

Then the dreaded Taylor Series formula tells us that

The problem with this formula is that it's impossible to picture an infinte number of functions added together. It doesn't seem like a simplification at all, just another big math headache. So, let's try and take each term in the series individually and see what happens.

Of course, the goal of this tutorial is to convince you that the Taylor Series is a useful tool to approximate your original function near your point. So, let's use two functions in this demonstration to illustrate the Taylor Series approximation:

a. Click HERE to see the graph of this function.

b. Click HERE to see the graph of this function.

The Order Approximation:

Okay, let's take just the first term in the Taylor Series, called the order approximation, and see what happens:

So, the Taylor Series approximation says that if I must approximate with a constant, the best approximation is the constant = . Well, that makes sense because the first term just evaluates the function at your given point.

To illustrate this point, take our first function, , and let . Then our order approximation is just .

Click HERE to see the graph of this constant function. For a closer look click HERE .

If you look closely at the point x = 1, you will notice how the value of becomes approximately .

Or, for , with , our order approximation is:

Click HERE to see the graph of this constant function around the point x = 0.

Look closely at the point x = 1. See how the value of is or ?

Notice how both functions gave us a mere constant when evaluated at the first term in the Taylor Series? In other words, you're simply finding the value of the function at the point . This isn't going to be very helpful if you want to see what happens not only on the point , but around the point . Even so, assuming we get the correct value at , if you had to approximate the function near by a constant, you can do no better than .

First Order Approximation:

The first order approximation takes the first two terms in the series and approximates the function a little further.

Okay, problem solving time. Let's compare the 2-term Taylor Series with the equation of the tangent line at . Recall from Calculus that the derivative gives the slope of the tangent line at a given point. So, using the point-slope form, find the equation of a straight line passing through the point with a slope of .

Point slope form: , where is a point on the line and m = slope. Since is on our line, and the slope of the tangent is , we get:

Recall:

So,

Look familiar? Yup, that's right! It's just our first two terms of the Taylor Series. The first order approximation is the equation of a line with a slope of . So, lo and behold, the first two terms of the Taylor Series gives us the equation of the line tangent to our function at the point . Well, that's the theory, but let's see what happens when we take the first order approximation of our two functions.

If you would like to see the graph of the order terms in the Taylor Series around the point when click HERE . If you have a graphing calculator handy, calculate the equation out for yourself and see if you come up with the same graph I did. You should get the equation evaluated at the point . This equals .

If you look closely (you are going to need a ruler for this), you can see that the Taylor Series approximated the function accurately within a certain interval. To find this interval, look at the graph to see approximately where the first order approximation overlapped the function, . I came up with the interval . To be consistent throughout this tutorial, we will define an interval where the truncated Taylor Series gives a good approximation for this function; so that the difference between the approximation and the original function is no more than throughout the interval.

If you would like to see the graph of the Order Approximation of evaluated at the point when click HERE . Go ahead and use your graphing calculator to produce the same graph. Your equation should be evaluated at the point .

Same as before, let's find the interval of accuracy for this function. I found it to be ; this time within an error of . Did you get the same thing? If you are having trouble finding the same interval, look at the graph and mark the interval I found. Then, take your ruler and measure the distance between the top function and the bottom function (our tangent line, ). This will give you the error.

Notice how the first two terms of the Taylor Series give us the equation of the line tangent to our function at the point ? This approximation works well, but only in a small region around our point . Of course, if the original function was fairly straight near , the first order approximation would be good over a larger interval. Let's see if we can use more terms of the series and find a better approximation of the function.

Second Order Approximation:

Congratulations! You made it through the first couple of calculations of the Taylor Series! Okay, let's go to the next step and see what happens. As you may have already guessed the second order approximation uses the first three terms of the Taylor Series.

Notice the term? This just happens to be the equation of a parabola. Okay, let's go ahead and derive this formula from the following information:

The equation of a parabola takes the form where we are using as our variable. Since we want the Taylor Approximation to equal the function at , and this implies , we require that:

. So, .

Now, since we want the parabola to have the same tangent line as the original function, let's take the first derivative of the function to find the slope of that tangent line:

. So, .

So far, all we have done is regenerate the first two terms in the Taylor Series. To go further, we need to require that the second derivatives agree. Imposing this condition gives:

. So, .

Hence, if you plug your constants A, B, and C back into your original function , you get the equation of the second order approximation of the Taylor Series.

Why might a parabola be a better approximation of our function? Well, our function is curved, and so is a parabola. If our function was straight, then the first two terms of the Taylor Series would approximate the function sufficiently on a larger interval. But since our function curves, the parabola will give us a chance for a better fit of our function.

Click HERE if you want to see what happens when we use the order approximation with the function at the same point . Have you calculated what the order approximation would be in this case? Go ahead and calculate the equation and graph it yourself to compare. The equation should be evaluated at the point . This gives us the equation .

Now we're getting somewhere! As you can see, the parabola produced by using the order approximation of the Taylor Series approximates the function within a region around more accurately than the tangent line. If you look closely, you can begin to see just how accurate the order approximation is compared to the function. The first three terms of the Taylor Series is approximating our function from about with error . Our previous interval was with error . Notice how our interval is getting larger with more terms of the Taylor Series?

Click HERE if you want to see what happens when we use the order approximation with the function at the point . The order approximation of this function becomes evaluated at the point . Remember: In order to get a better understanding of this concept, try and graph the function out for yourself.

The first three terms of the Taylor Series used with this function also produces a parabola that is a better approximation of our function around our region at . Notice how the order approximation tracks our original function around the interval with error ? Our old interval was with error . Do you suppose if we take more terms of the Taylor Series that we will get a better approximation of our function?

Third Order Approximation:

If you answered yes, then you were right! The order approximation takes the first four terms of the Taylor Series:

and approximates a function.

Click HERE if you want to see what happens when we use the order approximation with the function at the same point . This equation then becomes evaluated at the point . For a closer look at the graph, click HERE . Did you come out with the same graph? The order approximation is accurate within a certain interval. By looking at the graph, write down that interval.

The interval that I came up with was about with error . Did you come up with something similiar? Our interval using the Second Order Approximation was with error . In this approximation, the interval seems to get better in one direction and worse in the opposite. In general, we would expect the interval to get bigger as we increase the order. However, this is a good example why the remainder term in the Taylor Series is so important. We won't go into this right now, but keep it in mind. However, if you would like to solve this mystery now, Click HERE to see another tutorial that will explain this phenomenon. If you would like to hold off on learning why this happened right now, there will be an opportunity to click to the Remainder Term tutorial at the end of this one. For now, let's move on, okay?

Click HERE if you want to see what happens when we use the order approximation with the function at that same point . This equation is evaluated at the point .

The interval of accuracy is approximately with error . The interval using the second order approximation was with error . Go ahead and come up with the approximate interval of accuracy and compare to the interval I got. Once again the interval is getting larger using more terms of the Taylor Series. The Taylor Series is giving better approximations as you use more terms of the series.

Fourth Order Approximation:

Guess what? The fourth order approximation takes the first five terms in the Taylor Series and approximates the function even further!

Click HERE if you want to see what happens when we use the fourth order approximation with the function at the same point . The equation is evaluated at the point . Does your graph look the same?

The interval that I came up with was about with error . Did you come up with something similiar? Our last interval in the Order Approximation was with error . Now our interval is becoming larger with each term we add on to our approximation.

Click HERE if you want to see what happens when we use the fourth order approximation with the function at that same point. This equation is evaluated at the point .

Go ahead and come up with your interval. I got approximately with error . Our previous interval was with error . This just further proves our point.

What would happen if you went further, using more and more terms of the Taylor Series? If the trend from the first five terms of the Taylor Series continues, we expect that the approximation of the function would keep getting better and better. "With each addition of a term of higher degree, we get a polynomial with one more derivative that exactly matches the corresponding derivative of f at the special point of close fit . Thus, as the degree increases, we expect that the shapes of the polynomial graphs more and more resemble the shape of the graph of f." (Smith, pg. 610). However, sometimes the Taylor Series is only accurate around a fixed region. Beyond that region, the Taylor Series becomes inaccurate. That region, where the Taylor Series converges onto the original function is called the region within the radius of convergence. Let's call the radius of convergence R. Then the interval of convergence is defined to be .

The radius of convergence tells you how far from the fixed point the Taylor Series will converge onto the original function. On functions such as our example , you may not be surprised that the radius of convergence is large since the curve has a simple shape. In fact, you can notice from the graph, the Taylor Series seems to converge onto the entire function. In which case, the radius of convergence will be infinite and the interval of convergence is .

We use various techniques in Calculus to determine the interval of convergence. One of these techniques is called the ratio test. If is the term of the Taylor Series, the ratio test takes the where . Your function will converge for all values where this limit is less than one and diverge for all values greater than one. Let's use the ratio test to determine whether or not does indeed converge for all values of x. In this case .

The ratio test tells us:

Cancelling appropriate terms leaves:

for all h.

Therefore, we have proven that the function converges for all values of x. In fact, our function converges absolutely. When a series converges absolutely, the corresponding series of absolute values converges. Say what? This just means that if you take the absolute value of each of the terms in your series, it forms another series, called the corresponding series. If this corresponding series converges, then your original series converges absolutely. Our radius of convergence for this function is infinite. This is evident when we look at the graph of the Taylor Series at higher order terms. Our approximation becomes more accurate the more terms we use.

Perhaps not quite so obvious is our interval of convergence for our other function, . It's not easy to tell from the graph whether or not the radius of convergence is infinite. If we let .

Taking gives:

Cancelling terms leaves:

Assuming n is odd and or ,

and

Cancelling terms one more time leaves:

for all h.

Note 1: if n was even, the only difference in our final equation is a replacement of for . This would not change our result since the limit of this expression remains the same.

Note 2: if or , either the cosine or the sine term will equal 0. This also would not change our result since the limit of this expression remains the same. Thus, the radius of convergence of our function is infinite. If you would like to know more about the radius of convergence and the various techniques used, Click HERE to see a tutorial on the convergence of a infinite series. As promised, if you would like to explore the mystery of why the Order Approximation gave us unexpected behavior, click HERE to see a tutorial on the Remainder Term of the Taylor Series.

This tutorial was written by Jennifer J. Pearce in Spring, 1996 under the guidance of Prof. Kevin Short at the University of New Hampshire.

Notes:

Smith, David A., Lawrence C. Moore. Calculus Modeling and Applications. Copyright 1996 by D. C. Heath and Company.