Calculus: Early Transcendentals

James Stewart

Chapter 11 Infinite Sequences and Series - all with Video Answers

Educators

Section 8

Power Series

00:54

Problem 1

What is a power series?

Gabriel Rhodes

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02:10

Problem 2

(a) What is the radius of convergence of a power series?
How do you find it?
(b) What is the interval of convergence of a power series?
How do you find it?

Gabriel Rhodes

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01:30

Problem 3

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} ( - 1)^n nx^n $

Gabriel Rhodes

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02:13

Problem 4

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {( - 1)^n x^n}{\sqrt[3]{n}} $

Gabriel Rhodes

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02:16

Problem 5

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {x^n}{2n - 1} $

Gabriel Rhodes

Numerade Educator

01:51

Problem 6

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {( - 1)^n x^n}{n^2} $

Gabriel Rhodes

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01:12

Problem 7

Find the radius of convergence and interval of convergence of the series.

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

Gabriel Rhodes

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01:39

Problem 8

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} n^nx^n $

Gabriel Rhodes

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03:10

Problem 9

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {x^n}{n^44^n} $

Gabriel Rhodes

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03:07

Problem 10

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} 2^nn^2x^n $

Gabriel Rhodes

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03:18

Problem 11

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {( - 1)^n 4^n}{\sqrt{n}} x^n $

Gabriel Rhodes

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02:43

Problem 12

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n5^n} x^n $

Gabriel Rhodes

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04:08

Problem 13

Find the radius of convergence and interval of convergence of the series.

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

Gabriel Rhodes

Numerade Educator

03:14

Problem 14

Find the radius of convergence and interval of convergence of the series.

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

Gabriel Rhodes

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04:12

Problem 15

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 8}^{\infty} \frac {(x - 2)^n}{n^2 + 1} $

Gabriel Rhodes

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04:51

Problem 16

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {( - 1)^n}{(2n - 1)2^n} (x - 1)^n $

Gabriel Rhodes

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05:33

Problem 17

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 2}^{\infty} \frac {(x + 2)^n}{2^n \ln n} $

Gabriel Rhodes

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04:21

Problem 18

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {\sqrt{n}}{8^n} (x + 6)^n $

Gabriel Rhodes

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02:12

Problem 19

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {(x - 2)^n}{n^n} $

Gabriel Rhodes

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10:44

Problem 20

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {(2x - 1)^n}{5^ \sqrt{n}} $

Mutahar Mehkri

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04:26

Problem 21

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {n}{b^n} (x - a)^n, b > 0 $

Gabriel Rhodes

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05:39

Problem 22

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {b^n}{\ln n} (x - a)^n, b > 0 $

Gabriel Rhodes

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02:45

Problem 23

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} n!(2x - 1)^n $

Gabriel Rhodes

Numerade Educator

02:15

Problem 24

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {n^2x^n}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot \cdot (2n)} $

Gabriel Rhodes

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04:45

Problem 25

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {5x - 4)^n}{n^3} $

Gabriel Rhodes

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06:34

Problem 26

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 2}^{\infty} \frac {x^{2n}}{n(\ln n)^2} $

Gabriel Rhodes

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02:52

Problem 27

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot\cdot \cdot (2n - 1)} $

Gabriel Rhodes

Numerade Educator

06:34

Problem 28

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {n!x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)} $

Gabriel Rhodes

Numerade Educator

03:11

Problem 29

If $ \sum_{n = 0}^{\infty} c_n4^n $ is convergent, can we conclude that each of the following series is convergent?

(a) $ \sum_{n = 0}^{\infty} c_n ( - 2)^n $

(b) $ \sum_{n = 0}^{\infty} c_n ( - 4)^n $

Gabriel Rhodes

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01:37

Problem 30

Suppose that $ \sum_{n = 0}^{\infty} c_nx^n $ converges when $ x = - 4 $ and diverges when $ x = 6. $ What can be said about the convergence or divergence of the following series?

(a) $ \sum_{n = 0}^{\infty} c_n $

(b) $ \sum_{n = 0}^{\infty} c_n8^n $

(d) $ \sum_{n = 0}^{\infty} ( - 1)^n c_n 9^n $

Gabriel Rhodes

Numerade Educator

06:16

Problem 31

If $ k $ is a positive integer, find the radius of convergence of the series

$ \sum_{n = 0}^{\infty} \frac {(n!)^k}{(kn)!} x^n $

Gabriel Rhodes

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