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Shaun Ault

AP Calculus BC Review: Taylor and Maclaurin Series

Taylor and Maclaurin series are like polynomials, except that there are infinitely many terms. What makes these important is that they can often be used in place of other, more complicated functions. Read on to find out how!

Taylor and Maclaurin Polynomials

First of all, let’s recall Taylor Polynomials for a function f. You might want to read up on the topic here: AP Calculus BC Review: Taylor Polynomials

Assume n ≥ 0 is a fixed whole number. (This is the degree, or order, of the polynomial.)

Moreover, let’s assume c is a fixed real number (called the center).

Then the Taylor polynomial of f is:

Taylor polynomial definition

And the Maclaurin polynomial is just the Taylor polynomial centered at c = 0.

Maclaurin polynomial

Approximating Functions

Taylor and Maclaurin polynomials can approximate a function to any desired level of accuracy.

For example, see the figure below for graphs of the first few Maclaurin polynomials for f(x) = sin x.

Taylor polynomial approximations to a curve

The graphs of the Taylor polynomials of a function tend to match the original function’s graph closer and closer as the degree increases.

Taylor and Maclaurin Series

But Taylor and Maclaurin polynomials can only approximate functions. The approximation gets better and better with the inclusion of more terms. If you want an exact answer, then you have to include all the terms — all of the infinitely many terms!

Taylor series include all the terms!

So a Taylor series is really just taking a Taylor polynomial to its logical extreme — what happens when we let a Taylor polynomial’s degree increase without ever stopping?

Note, we still use the same formula for the Taylor coefficients as we did for Taylor polynomials:

Formula for the Taylor coefficients

(Remember, the superscript (n) means the nth derivative, not the nth power, of the function.)

Here’s the complete formula for the Taylor series of f, with center c (Taylor’s Formula):

Taylor series for a function f

The Maclaurin series is the same thing, but with c = 0 plugged in.

Convergence Issues

Because there are an infinite number of terms in a typical Taylor series, we have to address questions of convergence.

The convergence of a Taylor or Maclaurin series depends on the value of x. A given series will do one of three things:

  1. It may converge (have a finite value) regardless of x.
  2. There may be a finite number R (called the radius of convergence) so that the series converges within R units of the center c, and diverges outside of those bounds.
  3. The series may converge only at the center x = c, diverging at every other x-value.

But wait, it gets worse!

Sometimes, even if a Taylor series for f converges at some x-value, it may not actually have the same value as f(x)!

Don’t worry, though. On the AP Calculus BC exam, you will only see situations in which the Taylor series converges to the function within some finite radius or for all x.

In this article, we’ll just focus on producing Taylor and Maclaurin series, leaving their convergence properties to another post.

Differentiation and Integration

Taylor series are a type of power series. As such, you can do term-by-term differentiation and integration.

The following derivative and integral formulas apply to any power series — not just Taylor series.

Derivative Formula

derivative of a power series

Integral Formula

Integral of a power series

Common Power Series

Some functions are so common and useful that it just makes sense to memorize their power series.

Common Maclaurin series

There is no Maclaurin series for ln(x), because ln(0) is undefined. However, there’s a quick and easy fix. Just shift the function itself over by one unit.

Power series for ln(1+x)

Example: Hyperbolic Functions

The two functions, hyperbolic sine (sinh x) and hyperbolic cosine (cosh x) show up often in engineering.

Hyperbolic sine and cosine functions

Find the Maclaurin series for each one. Then I’ll show you two methods. You can pick which one seems better for you.

Solution #1

First, let’s see if we can use Taylor’s Formula.

We’ll need to know the derivatives of both sinh x and cosh x. Fortunately, there is a very simple pattern.

Derivative of sinh x

derivative of cosh x

Notice anything?

The derivative of sinh x is just cosh x, and vice versa!

Next, we’ll also need to know the values at c = 0.

values of sinh(0) = 0, and cosh(0) = 1

Now we have everything we need to put together the power series.

kf(k)(x)f(k)(1)/k!
0f(x) = sinh xsinh 0 = 0
1f '(x) = cosh xcosh 0 = 1
2f ''(x) = sinh x(sinh 0)/2! = 0
3f '''(x) = cosh x(cosh 0)/3! = 1/6
4f(4)(x) = sinh x(sinh 0)/4! = 0
5f(5)(x) = cosh x(cosh 0)/5! = 1/120

Only the odd degree terms are nonzero. Here’s what it looks like when you put it all together.

power series for sinh x

The power series for cosh x follows much the same pattern, except that now it’s the even degree terms that are nonzero.

power series for cosh x

Solution #2

Now let’s use the known power (Maclaurin) series for ex to derive a power series for sinh x and cosh x.

Remember: power series for e^x.

Then, in the formula for sinh x, you can just use the series for ex everywhere you see an “ex.”

Showing the steps to work out sinh x as a series

That last line may be simplified further. There is a pattern at work here.

  • If n is even, then 1 – (-1)n – 1 – 1 = 0.
  • But if n is odd, then 1 – (-1)n – 1 – (-1) = 2.

So actually, only the odd degree terms survive in the sum. (Sound familiar?)

And in that case, the 1/2 cancels with the 2, so that only the power of x and the factorial remains in each term.

Steps for producing the Maclaurin series for sinh x

Finally, by re-indexing the sum in terms of odd whole numbers (2n + 1), we get the proper Maclaurin series for sinh x.

power series for sinh x

You can do the same sort of work to derive the series for cosh x. The first few steps are shown below, but the result will be the same series that we got before for hyperbolic cosine.

Working out the cosh x series

Graphs of Hyperbolic Functions

Finally, to round out this example problem, let’s take a look at the graphs of each hyperbolic function. It would be a great exercise for you to plot the first few Taylor polynomials of each series and compare the results against these graphs.

Graph of hyperbolic sine

Graph of y = sinh x from Wolfram MathWorld.

Graph of cosh x

Graph of y = cosh x from Wolfram MathWorld.

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About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!


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