TBIL-Calc Convergence of Power Series (PS2)

Section 9.2 Convergence of Power Series (PS2)

Learning Outcomes

Determine the interval of convergence for a given power series.

Remark 9.2.1.

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}

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Activity 9.2.2.

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

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(a)

Letting $c_{n} = \frac{1}{n^{2} + 1},$ find $lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | .$

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(b)

For what values of $x$ is $| x | lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | < 1 ?$

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

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(c)

If $x = 1,$ does $\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n}$ converge?

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(d)

If $x = - 1,$ does $\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n}$ converge?

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(e)

Which of the following describe the values of $x$ for which $\sum_{n = 0}^{\infty} \frac{1}{n^{2} + 1} x^{n}$ converges?

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

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Activity 9.2.3.

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

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(a)

Letting $c_{n} = \frac{2^{n}}{5^{n}},$ find $lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | .$

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(b)

For what values of $x$ is $| x - 2 | lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | < 1 ?$

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

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(c)

If $x = \frac{9}{2},$ does $\sum_{n = 0}^{\infty} \frac{2^{n}}{5^{n}} (x - 2)^{n}$ converge?

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(d)

If $x = - \frac{1}{2},$ does $\sum_{n = 0}^{\infty} \frac{2^{n}}{5^{n}} (x - 2)^{n}$ converge?

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(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?

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

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Activity 9.2.4.

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

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(a)

Letting $c_{n} = \frac{n^{2}}{n!},$ find $lim_{n \to \infty} | \frac{c_{n + 1}}{c_{n}} | .$

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(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.

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(c)

What describes the values of $x$ for which $\sum_{n = 0}^{\infty} \frac{n^{2}}{n!} {(x + \frac{1}{2})}^{n}$ converges?

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Fact 9.2.5.

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.

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Activity 9.2.6.

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!} .

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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} {(x + 2)}^{n}}{n} .

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Activity 9.2.8.

Consider the power series $\sum_{n = 0}^{\infty} \frac{2^{n} + 1}{n 3^{n}} {(x + 1)}^{n} .$

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(a)

What is the center of convergence for this power series?

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(b)

What is the radius of convergence for this power series?

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(c)

What is the interval of convergence for this power series?

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(d)

If $x = - 0.5,$ does this series converge? (Use the interval of convergence.)

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(e)

If $x = 1,$ does this series converge? (Use the interval of convergence.)

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Subsection 9.2.1 Videos

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Figure 168. Video: Determine the interval of convergence for a given power series