1231 Taylor Power Ser 2010

MATH 1231 MATHEMATICS 1B 2010.For use in Dr Chris Tisdell’s lectures.

Calculus Section 4.4: Taylor & Power series.

1. What is a Taylor series?

2. Convergence of Taylor series

3. Common Maclaurin series

4. Applications

5. What is a power series?

6. Motivation

7. Radius of convergence

8. Manipulation of power series

Lecture notes created by Chris Tisdell. All images are from “Thomas’ Calcu-lus” by Wier, Hass and Giordano, Pearson, 2008; and “Calculus” by Rogowski,W H Freeman & Co., 2008.

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1. What is a Taylor series?

We now extend the idea of Taylor polynomials to Tay-lor series.

In particular, we can now fulfil our aim of developing a method for representing

a (differentiable) function f(x) as an (infinite) sum of powers of x. The main

thought–process behind our method is that powers of x are easy to evaluate,

differentiate and integrate, so by rewriting complicated functions as sums of

powers of x we can greatly simplify our analysis.

Colin Maclaurin was a professor of mathematics at Edinburgh univer-

sity. Newton was so impressed by Maclaurin’s work that he offered to pay part

of Maclaurin’s salary.

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Ex. Find the Maclaurin series for sinx.

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Ex. Find the Maclaurin series for cosx.

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Ex. Find the Maclaurin series for log(1 + x).

What happens if we try to find the Maclaurin series forlogx in powers of x?

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Ex. Find the Taylor series for the function logx aboutthe point x = 1.

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2. Convergence.

Recall from the theory of Taylor polynomials that Rk(x),the remainder of order k, between a function f(x) andits Taylor polynomial Pk(x) of order k is just

Rk(x) = f(x)− Pk(x).

Since

P (x) :=∞∑

n=0

f(n)(a)

n!(x− a)n

is the limit of the partial sums Pk(x) we see that theTaylor series converges to f(x) if and only if

limk→∞

Rk(x) = 0. (1)

The following theorem gives us a method of determin-ing when (1) holds.

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Ex: Find the Taylor series for the function cosx aboutthe point x = π

2, (i.e. in powers of (x− π2).) On what

interval does the Taylor series converge to cosx?

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3. Common Maclaurin series.

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4. Applications.

This section contains some nice applications of Taylorseries. We look at limits with indeterminate forms.

Ex. Evaluate limx→0sinx

x .

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We now move onto definite integrals where the inte-grand does not have an explicit antiderivative.

Ex. Express I :=∫ 10 sin(x2) dx as an infinite series.

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5. What is a power series?

We have already seen two special power series:

the Maclaurin series for a given function f(x) is apower series about x = 0 with

an =f(n)(0)

n!in (1);

the Taylor series for a given function f(x) about x = a

is a power series about x = a with

an =f(n)(c)

n!in (2).

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6. Motivation

Similar to polynomials, convergent power series canbe added, subtracted, multiplied, differentiated and in-tegrated to form new power series.

Consider the function

F (x) =∞∑

n=0

bnxn =∞∑

n=0

1

2nxn.

For what values of x does the above series converge?That is, what is the domain of F?It is clear that F (0) is defined, since substitution into the seriesgives

F (0) = 1 + 0 + 0 + . . . = 1.

What about F (1)? Substitution into the series yields

F (1) =∞∑

n=0

1

2n= 1 + 1/2 + 1/4 + 1/8 + . . . = 2.

What about F (3)? We see

F (3) =∞∑

n=0

1

2n3n

which is a geometric series that diverges. Thus, F (3) is notdefined.

Can we determine all values of x without substitu-tion??

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7. Radius of convergence.

To determine the radius of convergence, we can em-ploy the ratio test from our earlier work on infinite se-ries.

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Let us return to our previous example of

F (x) =∞∑

n=0

bnxn =∞∑

n=0

1

2nxn.

We compute the following ratio ρ, applying the ratio test

ρ = limn→∞

∣∣∣∣bn+1xn+1

bnxn

∣∣∣∣= lim

n→∞

∣∣∣∣xn+1

2n+1

∣∣∣∣ ∣∣∣∣2n

xn

∣∣∣∣=

1

2limn→∞

|x|

=1

2|x|.

By the ratio test, our series will converge when ρ = |x|/2 < 1.

That is, when |x| < 2. Similarly, our series will diverge when

ρ > 1, that is, when |x| > 2.

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Ex. Discuss the convergence of the power series

F (x) =∞∑

n=0

xn

n!.

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Ex. Discuss the convergence of

F (x) =∞∑

n=1

(−1)n

n(x− 5)n.

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Ex. Discuss the convergence of

F (x) =∞∑

n=1

nxn.

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8. Manipulation of power series.

The following theorems state that a power series F (x) can be

differentiated and integrated within its interval of convergence.

We may differentiate and integrate F (x) as if it were a polyno-

mial.

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Ex: Establish a power series for log(1 + x), −1 <

x < 1 by using the power series∞∑

n=0

tn, −1 < t < 1.

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