Calculus II
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Taylor and Maclaurin series

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01 Theory

Theory 1

Suppose that we have a power series function:

Consider the successive derivatives of :

When these functions are evaluated at , all terms with a positive -power become zero:

This last formula is the basis for Taylor and Maclaurin series:

Power series: Derivative-Coefficient Identity

This identity holds for a power series function which has a nonzero radius of convergence.

We can apply the identity in both directions:

  • Know ? Calculate for any .
  • Know ? Calculate for large . (Faster than differentiating.)

Many functions can be ‘expressed’ or ‘represented’ near (i.e. for small enough ) as convergent power series. (This is true for almost all the functions encountered in pre-calculus and calculus.)

Such a power series representation is called a Taylor series. When , the Taylor series is also called the Maclaurin series.

One power series representation we have already studied:

Whenever a function has a power series (Taylor or Maclaurin), the Derivative-Coefficient Identity may be applied to calculate the coefficients of that series.

Conversely, sometimes a series can be interpreted as an evaluated power series coming from for some . If the closed form function format can be obtained for this power series, the total sum of the original series may be discovered by putting in the argument of the function.

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02 Illustration

Example - Maclaurin series of

Maclaurin series of e to the x

What is the Maclaurin series of ?

Solution

Using repeatedly, we see that for all .

So for all . Therefore for all by the Derivative-Coefficient Identity:

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Example - Maclaurin series of

Maclaurin series of cos x

Find the Maclaurin series representation of .

Solution

Use the Derivative-Coefficient Identity to solve for the coefficients:

0
1
2
3
4
5

By studying this pattern, we find the series:

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Maclaurin series from other Maclaurin series

Maclaurin series from other Maclaurin series

(a) Find the Maclaurin series of using the Maclaurin series of .

(b) Find the Maclaurin series of using the Maclaurin series of .

(c) Using (b), find the value of .

Solution

(a)

Remember that . Let us differentiate the cosine series by terms:

Take negative to get:

(b)

Set :

Multiply all terms by :

(c)

For any series:

we have:

We can use this to compute . From the series formula:

we see that:

Power, NOT term number

The coefficient with corresponds to the term having , not necessarily the term of the series.

Therefore:

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Computing a Taylor series

Computing a Taylor series

Find the first five terms of the Taylor series of centered at .

Solution

A Taylor series is just a Maclaurin series centered at a nonzero number.

General format of a Taylor series:

The coefficients satisfy .

Find the coefficients by computing the derivatives and evaluating at :

The first terms of the series:

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03 Theory

Theory 2

Study these!

  • Memorize all of these series!
  • Recognize all of these series!
  • Recognize all of these summation formulas!
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Applications of Taylor series

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04 Theory reminder

Theory 1

Linear approximation is the technique of approximating a specific value of a function, say , at a point that is close to another point where we know the exact value . We write for , and , and . Then we write and use the fact that:

Computing a linear approximation

For example, to approximate the value of , set , set and , and set so .

Then compute: So .

Finally:

Now recall the linearization of a function, which is itself another function:

Given a function , the linearization at the basepoint is the functional form of the tangent line, the line passing through with slope :

The graph of this linearization is the tangent line to the curve at the point .

The linearization may be used as a replacement for for values of near . The closer is to , the more accurate the approximation is for .

Computing a linearization

We set , and we let .

We compute , and so .

Plug everything in to find :

Now approximate :

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05 Theory

Theory 2

Taylor polynomials

The Taylor polynomials of a function are the partial sums of the Taylor series of :

These polynomials are generalizations of linearization. Specifically, , and .

The Taylor series is a better approximation of than for any .

center

center

Facts about Taylor series

The series has the same derivatives as at the point . This fact can be verified by visual inspection of the series: apply the power rule and chain rule, then plug in and all factors left with will become zero.

The difference vanishes to order at :

The factor drives the whole function to zero with order as .

If we only considered orders up to , we might say that and are the same near .

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06 Illustration

Taylor polynomial approximations

Taylor polynomial approximations

Let and let be the Taylor polynomials expanded around .

By considering the alternating series error bound, find the first for which must have error less than .

Solution

Write the Maclaurin series of because we are expanding around :

This series is alternating, so the AST error bound formula applies (“Next Term Bound”):

Find smallest such that , and then we know:

Plug in the series for :

Solve for the first time by listing the values:

The first time is below happens when .

This is NOT the same as in . That is the highest power of allowed.

The sum of prior terms is .

Since because there is no term, the final answer is .

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Taylor polynomials to approximate a definite integral

Taylor polynomials to approximate a definite integral

Approximate using a Taylor polynomial with an error no greater than .

Solution

Plug into the series of :

Find an antiderivative by terms:

Plug in bounds for definite integral:

Notice alternating series pattern. Apply error bound formula, “Next Term Bound”:

So we can guarantee an error less than by summing the first terms through :

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