TBIL-Cal Power Series (PS1)

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Section 9.1 Power Series (PS1)

Learning Outcomes

Approximate functions defined as power series.

Subsection 9.1.1 Activities

Activity 9.1.1.

Suppose we could define a function as an “infinite-length polynomial”:

f (x) = 1 + x + x^{2} + x^{3} + x^{4} + \dots .

(a)

Would

f (1)

be well-defined as a finite real number?

No, the sum would diverge towards $\infty .$
No, the sum would oscillate between $0$ and $1 .$
Yes, the sum would be $0 .$
Yes, the sum would be $1 .$

(b)

Would

f (- 1)

be well-defined as a finite real number?

No, the sum would diverge towards $\infty .$
No, the sum would oscillate between $0$ and $1 .$
Yes, the sum would be $0 .$
Yes, the sum would be $1 .$

(c)

Would

f (1 / 2)

be well-defined as a finite real number?

No, the sum would diverge towards $\infty .$
Yes, the sum would be approximately $1 .$
Yes, the sum would be approximately $2 .$
Yes, the sum would be exactly $2 .$

(d)

When is

f (x)

well-defined as a finite real number?

Its value is $\frac{x}{1 - x}$ when $| x | < 1 .$
Its value is $\frac{x}{1 - x}$ when $x < 1 .$
Its value is $\frac{1}{1 - x}$ when $| x | < 1 .$
Its value is $\frac{1}{1 - x}$ when $x < 1 .$

Definition 9.1.2.

Given a sequence of numbers

a_{n}

and a number

c,

we may define a function

f (x)

as a power series:

f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + a_{3} (x - c)^{3} + \dots .

The above power series is said to be centered at

c

. Often power series are centered at

0;

in this case, they may be written as:

f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots .

The domain of this function (often referred to as the domain of convergence or interval of convergence) is exactly the set of

x

-values for which the series converges.

Activity 9.1.3.

In Section 9.2 we will learn how to prove that

\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}

converges for each real value

x .

Thus the function

f (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120} + \dots

has the domain of all real numbers.

(a)

To estimate

f (2),

use technology to compute the first few terms as follows:

\begin{aligned} f (2) = \sum_{n = 0}^{\infty} \frac{2^{n}}{n!} & = 1 + 2 + \frac{2^{2}}{2} + \frac{2^{3}}{6} + \frac{2^{4}}{24} + \frac{2^{5}}{120} + \dots \\ = ? + \dots \\ \approx ? \end{aligned}

Which of these choices is the closest to this value?

$\sqrt{2} \approx 1.414 .$
$e^{2} \approx 7.389 .$
$\sin (2) \approx 0.909 .$
$\cos (2) \approx - 0.416 .$

(b)

Estimate

f (- 1)

in a similar fashion:

\begin{aligned} f (- 1) = \sum_{n = 0}^{\infty} \frac{?}{n!} & = ? + ? + ? + ? + ? + ? + \dots \\ = ? + \dots \\ \approx ? \end{aligned}

Which of these choices is the closest to this value?

$\frac{1}{\sqrt{1}} \approx 1.000 .$
$\frac{1}{e^{1}} \approx 0.369 .$
$\frac{1}{\sin (1)} \approx 1.188 .$
$\frac{1}{\cos (1)} \approx 1.851 .$

Activity 9.1.4.

The function

f (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = \sum_{n = 0}^{\infty} \frac{1}{n!} (x - 0)^{n}

is centered at

0 .

Likewise, graphing the polynomial that uses the first six terms

f_{5} (x) = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120}

alongside the graph of

e^{x}

reveals the illustration given in the following figure.

described in detail following the image — Figure 192. Plots of $y = f_{5} (x), y = e^{x} .$

What might we conclude?

$e^{x} \approx 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120}$ near $x = 0 .$
$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120}$ near $x = 0 .$
$e^{x} \approx 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120}$ for all $x .$
$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120}$ for all $x .$

Definition 9.1.5.

Given a power series

f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + a_{3} (x - c)^{3} + \dots,

let

f_{N} (x) = \sum_{n = 0}^{N} a_{n} (x - c)^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + \dots + a_{N} (x - c)^{N}

be its degree

N

polynomial approximation for

x

nearby

c .

For example,

\begin{aligned} g_{3} (x) = \sum_{n = 0}^{3} n^{2} (x - 1)^{n} & = 0 + (x - 1) + 4 (x - 1)^{2} + 9 (x - 1)^{3} \\ = - 6 + 20 x - 23 x^{3} + 9 x^{3} \end{aligned}

is a degree

3

approximation of

g (x) = \sum_{n = 0}^{\infty} n^{2} (x - 1)^{n}

valid for

x

values nearby

1 .

Activity 9.1.6.

Consider a function

p (x)

defined by

p (x) = \sum_{n = 0}^{\infty} \frac{2^{n}}{(2 n)!} x^{n} .

(a)

Find

p_{3} (x),

the degree 3 polynomial approximation for

p (x) .

(b)

Use

p_{3} (x)

to estimate

p (- 1) .

Subsection 9.1.2 Videos

Figure 193. Video: Approximate functions defined as power series

Subsection 9.1.3 Exercises

Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/PS1/