Infinite Series: Part 2

Previously we have laid the foundation for understanding infinite series. We have seen how to find the sum of an infinite series, and how to determine whether a series converges or diverges using different tests. Now we will explore some more advanced topics related to infinite series.

Table of Contents

• Table of Contents
• Taylor and Maclaurin Series
  • Relationship Between Taylor Series and the Fundamental Theorem of Calculus
  • Example Problem: Finding the n-th Maclaurin Polynomial of a Function

Taylor and Maclaurin Series

The Taylor series is an extremely powerful tool in mathematics, one of the best ways to approximate functions. It allows for the approximation of a non-polynomial function using a polynomial. Polynomials are easy to compute, manipulate, differentiate, and integrate, so approximating a function with a polynomial can make it much easier to work with.

To start, consider a function . Assume this is part of some physics problem, and the existence of a term is making the problem difficult to solve. Let's say we are interested in the value of near , and we want to approximate it with a degree 2 polynomial (a quadratic function).

Denote the quadratic function as . Play around with the sliders below to see how the quadratic function changes as you adjust the coefficients a_0, a_1, and a_2:

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Consider how the quadratic function changes as you adjust the coefficients , , and . The goal is to find the coefficients that make as close as possible to near .

First, consider evaluated at :

We want the value of to be the same as at .

Hence, one of the coefficients is already determined. Changing the other coefficients will affect the function but will not change the value of .

Next, we want the slope of to be the same as the slope of at . This ensures that the value of is close to not only at but also in the neighborhood of ; it doesn't drift too far away from .

Now, both the position and the slope of are locked as the same as at . The only thing left is to make sure the concavity of is the same as the concavity of at .

Now we have found the coefficients , , and . The quadratic functoin is then:

The quadratic function is a good approximation of near . As we add more terms to the polynomial, we can get even closer to the original function. For example, we can add two more terms to get a degree 4 polynomial:

The coefficients and can be determined in a similar way as we did for . We take the third and fourth derivatives of at and set them equal to the corresponding derivatives of at . Notice what happens when we take the third derivative of the term in :

Notice we are multiplying by by by . This is the same as taking the factorial of :

Recall that we want this to be equal to the third derivative of at . Thus:

And as such, we can solve for :

We get a similar result for :

Thus, the degree 4 polynomial approximation of is:

Notice that adding the higher degree term does not affect the previous terms. For instance, the concavity of is still the same as the concavity of at .

Why?

When we take the derivative of a term with a higher power of , the power of decreases by one. The concavity is the second derivative, so the power of decreases by two - so it doesn't become a constant term.

Recall that we set . As such, if there is an term in the polynomial, it will be multiplied by zero, and the term will disappear.

Hence, each individual coefficient is the "control" for a specific order derivative of the function.

For a value of near , the polynomial is a good approximation of . Next, consider approximating with a polynomial near some value .

Then, the polynomial would be written as:

Instead of , we use as the variable, such that the point is treated like . This means that if you plug in , things will cancel.

We can keep adding more terms to the polynomial to get a better approximation of near a value of . The derivatives of follow a pattern of at . As such, the coefficients of the polynomial will follow a similar pattern:

Next, we can generalize this to any function :

And to any value :

We have just derived the Taylor series for any function centered at . This can be written in sigma notation as:

When , this is called the Maclaurin series.

What's interesting is that we only need to know the value of the function and its derivatives at a single point to approximate the function at any other point around it.

Relationship Between Taylor Series and the Fundamental Theorem of Calculus

There's a deep connection between the Taylor series and the Fundamental Theorem of Calculus. Recall one interpretation of the Fundamental Theorem of Calculus, which is that the derivative of the area under a curve is the curve itself.

Consider the area under the curve of a function from to some variable :

Let the area under the curve be .

Next, consider adding a small change to the variable that increases to :

The total area can be approximated by the blue, green, and red areas.

• The blue area is the current area under the curve, .
• The green area is a rectangle;
  • The width is the change in , .
  • The height is the value of the function at , , which is the slope of the area .
  • Hence the area of the green rectangle is .
• The red area is a small triangle;
  • The base is the change in , .
  • The height is the slope of the function at , . Since is the derivative of , this is the second derivative of .
  • Hence the area of the red triangle is .

The total area is then:

This is equivalent to the Taylor series expansion of centered at :

This shows how interconnected the various concepts in calculus are.

Example Problem: Finding the n-th Maclaurin Polynomial of a Function

Consider the function defined as follows:

Identify its second degree Maclaurin polynomial.

(Source)

Recall from Equation that the Maclaurin series of a function centered at is:

We only need to find the second degree Maclaurin polynomial, so we only need to find the first three terms of the series:

The derivatives of are:

Evaluating these derivatives at :

Substitute these values into the polynomial: