TBIL-Cal Convergence of Power Series (PS2)

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Section 9.2 Convergence of Power Series (PS2)

Learning Outcomes

Determine the interval of convergence for a given power series.

Subsection 9.2.1 Activities

Activity 9.2.1.

Consider the series

\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n}

where

x

is a real number.

(a)

If

x = 2,

then

\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{2^{n}}{n!} .

What can be said about this series?

The techniques we have learned so far allow us to conclude that $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{2^{n}}{n!}$ converges.
The techniques we have learned so far allow us to conclude that $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{2^{n}}{n!}$ diverges.
None of the techniques we have learned so far allow us to conclude whether $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{2^{n}}{n!}$ converges or diverges.

(b)

If

x = - 100,

then

\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{(- 100)^{n}}{n!} .

What can be said about this series?

The techniques we have learned so far allow us to conclude that $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{(- 100)^{n}}{n!}$ converges.
The techniques we have learned so far allow us to conclude that $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{(- 100)^{n}}{n!}$ diverges.
None of the techniques we have learned so far allow us to conclude whether $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{(- 100)^{n}}{n!}$ converges or diverges.

(c)

Suppose that

x

were some arbitrary real number. What can be said about this series?

The techniques we have learned so far allow us to conclude that $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n}$ converges.
The techniques we have learned so far allow us to conclude that $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n}$ diverges.
None of the techniques we have learned so far allow us to conclude whether $\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n}$ converges or diverges.

Remark 9.2.2.

Consider a power series

\sum c_{n} (x - a)^{n} .

Recall from Fact 8.7.6 that if

\begin{aligned} lim_{n \to \infty} | \frac{c_{n + 1} (x - a)^{n + 1}}{c_{n} (x - a)^{n}} | & < 1 \end{aligned}

then

\sum c_{n} (x - a)^{n}

converges.

Then recall:

\begin{aligned} lim_{n \to \infty} | \frac{c_{n + 1} (x - a)^{n + 1}}{c_{n} (x - a)^{n}} | & = lim_{n \to \infty} | \frac{c_{n + 1} (x - a)}{c_{n}} | \\ = lim_{n \to \infty} | x - a | | \frac{c_{n + 1}}{c_{n}} | \\ = | x - a | lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | . \end{aligned}

Activity 9.2.3.

Consider

\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n} .

(a)

Letting

c_{n} = \frac{1}{n^{2} + 1},

find

lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | .

(b)

For what values of

x

is

| x | lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | < 1 ?

$x < 1 .$
$0 \leq x < 1 .$
$- 1 < x < 1 .$

(c)

If

x = 1,

does

\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n}

converge?

(d)

If

x = - 1,

does

\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n}

converge?

(e)

Which of the following describe the values of

x

for which

\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n}

converges?

$(- 1, 1) .$
$[- 1, 1) .$
$(- 1, 1] .$
$[- 1, 1] .$

Activity 9.2.4.

Consider

\sum_{n = 0}^{\infty} \frac{2^{n}}{5^{n}} (x - 2)^{n} .

(a)

Letting

c_{n} = \frac{2^{n}}{5^{n}},

find

lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | .

(b)

For what values of

x

is

| x - 2 | lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | < 1 ?

$- \frac{2}{5} < x < \frac{2}{5} .$
$\frac{8}{5} < x < \frac{12}{5} .$
$- \frac{5}{2} < x < \frac{5}{2} .$
$- \frac{1}{2} < x < \frac{9}{2} .$

(c)

If

x = \frac{9}{2},

does

\sum_{n = 0}^{\infty} \frac{2^{n}}{5^{n}} (x - 2)^{n}

converge?

(d)

If

x = - \frac{1}{2},

does

\sum_{n = 0}^{\infty} \frac{2^{n}}{5^{n}} (x - 2)^{n}

converge?

(e)

Which of the following describe the values of

x

for which

\sum_{n = 0}^{\infty} \frac{2^{n}}{5^{n}} (x - 2)^{n}

converges?

$(- \frac{1}{2}, \frac{9}{2}) .$
$[- \frac{1}{2}, \frac{9}{2}) .$
$(- \frac{1}{2}, \frac{9}{2}] .$
$[- \frac{1}{2}, \frac{9}{2}] .$

Activity 9.2.5.

Consider

\sum_{n = 0}^{\infty} \frac{n^{2}}{n!} {(x + \frac{1}{2})}^{n} .

(a)

Letting

c_{n} = \frac{n^{2}}{n!},

find

lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | .

(b)

For what values of

x

is

| x + \frac{1}{2} | lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | < 1 ?

$0 \leq x < \infty .$
All real numbers.

(c)

What describes the values of

x

for which

\sum_{n = 0}^{\infty} \frac{n^{2}}{n!} {(x + \frac{1}{2})}^{n}

converges?

Fact 9.2.6.

Given the power series

\sum c_{n} (x - a)^{n},

the center of convergence is

x = a .

The radius of convergence is

r = \frac{1}{lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} |} .

If

lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | = 0,

we say that

r = \infty .

The interval of convergence represents all possible values of

x

for which

\sum c_{n} (x - a)^{n}

converges, which is of the form:

$(a - r, a + r)$
$[a - r, a + r)$
$(a - r, a + r]$
$[a - r, a + r]$

Depending on if

\sum c_{n} (x - a)^{n}

converges when

x = a - r

or

x = a + r .

If

r = \infty,

the interval of convergence is all real numbers.

Activity 9.2.7.

Find the center of convergence, radius of convergence, and interval of convergence for the series:

\sum_{n = 0}^{\infty} \frac{3^{n} {(- 1)}^{n} {(x - 1)}^{n}}{n!} .

Activity 9.2.8.

Find the center of convergence, radius of convergence, and interval of convergence for the series:

\sum_{n = 0}^{\infty} \frac{3^{n} {(x + 2)}^{n}}{n} .

Activity 9.2.9.

Consider the power series

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

(a)

What is the center of convergence for this power series?

(b)

What is the radius of convergence for this power series?

(c)

What is the interval of convergence for this power series?

(d)

If

x = - 0.5,

does this series converge? (Use the interval of convergence.)

(e)

If

x = 1,

does this series converge? (Use the interval of convergence.)

Subsection 9.2.2 Videos

Figure 189. Video: Determine the interval of convergence for a given power series

Subsection 9.2.3 Exercises

Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/PS2/