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Calculus BC Power Series Convergence

Section 5: Sequences, Series, & Approximations: Lecture 10 | 12:52 min

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Lecture Comments (9)

	1 answer Last reply by: Chrystal Wang Mon Apr 15, 2019 11:45 PM Post by Jorge Sardinas on June 3, 2014 In the 3 rd example he says that it is x^(n+1) when it is SUPPOSED to be x^(n-1).
	1 answer Last reply by: John Zhu Tue Aug 13, 2013 8:30 PM Post by Timothy Davis on August 5, 2013 How did u make the x -1 go away. U mentioned something about "rational rules. This was around 5:15 into your lecture on Power Series example 2. Somehow (x -1) magically just disappeared!
	0 answers Post by Chung Teak Joon on May 1, 2012 Example 1 seems to be wrong to me....because in the previous video, you said (an) convergies if L is smaller than zero..... I think you did it in the other way, or the ratio test video's content is simply wrong.
	1 answer Last reply by: Yiru Shi Wed Feb 10, 2016 2:44 PM Post by robin joshi on April 25, 2012 The video is absolutely wrong here!!
	1 answer Last reply by: John Zhu Mon Aug 12, 2013 9:26 PM Post by William Hamilton on April 19, 2012 When L<1, the series DOES converge absolutely. I think you had it backwards here.

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Power Series Convergence

Power series:
Power series can be convergence on an interval of x
Test for power series convergence:

Use ratio test to check for absolute convergence
If not absolutely convergent, solve for x value when it is absolutely convergent
Check x bounds for convergence by plugging in x to create series

Power Series Convergence

Find the values of x such that ∑_{n = 1}^∞ [((x − 2)ⁿ)/n] converges

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [((x − 2)^{n + 1})/(n + 1)]( [n/((x − 2)ⁿ)] ) | = lim_{n →∞} | [((x − 2)n)/(n + 1)] |[n/(n + 1)][n/n] = 1

| x − 2 | < 1 − 1 < x − 2 < 11 < x < 3

Find the values of x such that ∑_{n = 1}^∞ [((x + 1)ⁿ)/(n²)] converges

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [((x + 1)^{n + 1})/((n + 1)²)]( [(n²)/((x + 1)ⁿ)] ) | = lim_{n →∞} | [((x + 1)n²)/(( n + 1 )²)] |[(n²)/(( n + 1 )²)][(n²)/(n²)] = 1

| x + 1 | < 1 − 1 < x + 1 < 1 − 2 < x < 0

Find a value of x such that ∑_{n = 1}^∞ [((3x² − 1)ⁿ)/(n^3/2)] converges

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [((3x² − 1)^{n + 1})/((n + 1)^3/2)]( [(n^3/2)/((3x² − 1)ⁿ)] ) | = lim_{n →∞} | [((3x² − 1)n^3/2)/(( n + 1 )^3/2)] |[(n^3/2)/(( n + 1 )^3/2)][(n^3/2)/(n^3/2)] = 1
Find the domain of convergence
| 3x² − 1 | < 1

0 < 3x²< 20 < x²< [2/3]0 < x <√{[2/3]}

Find the values of x such that ∑_{n = 1}^∞ [n/((x − 1)ⁿ)] converges

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [(n + 1)/((x − 1)^{n + 1})]( [((x − 1)ⁿ)/n] ) | = lim_{n →∞} | [(n + 1)/((x − 1)n)] |[(n + 1)/n][n/n] = 1
Find the domain of convergence
[1/(| x − 1 |)] < 1
[1/(x − 1)] < 1x > 2

[1/(x − 1)] >− 1x < 0

Determine if the series ∑_{n = 1}^∞ [(xⁿ)/(eⁿ)] converges absolutely, and if so what's the domain of x

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [((x)^{n + 1})/(e^{n + 1})]( [(eⁿ)/(xⁿ)] ) | = lim_{n →∞} | [x/e] |[x/e]

| [x/e] | < 1 − 1 < [x/e] < 1 − e < x < e

Determine if the series ∑_{n = 1}^∞ [((x + 1)ⁿ)/n!] converges absolutely

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [((x + 1)^{n + 1})/((n + 1)!)]( [n!/((x + 1)ⁿ)] ) | = lim_{n →∞} | [(x + 1)/(( n + 1 ))] | = 0 < 1 The series converges, but does not converge absolutely

Find a value of x such that ∑_{n = 1}^∞ [(( − 1)ⁿ)/(n + 3)] converges

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [(x^{n + 1})/(n + 4)]( [(n + 3)/(xⁿ)] ) | = lim_{n →∞} | x[(n + 3)/(n + 4)] ||x|< 1
Find the domain of convergence
| x | < 1
− 1 < x < 1

Test values
Note that if x = − 1, the series converges via Alternate Series Test. Thus the range is actually
− 1 ≤ x < 1

Find the values of x such that ∑_{n = 1}^∞ [(eⁿ)/((x − 12)ⁿ)] converges

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [(e^{n + 1})/((x − 12)^{n + 1})]( [((x − 12)ⁿ)/(eⁿ)] ) | = lim_{n →∞} | [e/((x − 12))] | < 1| [e/((x − 12))] | < 1
Find the domain of convergence
| [e/((x − 12))] | < 1
[e/((x − 12))] < 1e + 12 < x

[e/((x − 12))] >− 112 − e > x

Determine if the series ∑_{n = 1}^∞ [(xⁿ)/n!] converges absolutely, and if so what's the domain of x

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [((x)^{n + 1})/((n + 1)!)]( [n!/(xⁿ)] ) | = lim_{n →∞} | [x/(n + 1)] |0 < 1 The series converges, but does not converge absolutely

Determine if the series ∑_{n = 1}^∞ [(( − 1)ⁿxⁿ)/(n²)] converges absolutely if so what's the domain of x

Apply the Ratio Test
lim_{n →∞} | [(a_{n + 1})/(a_n)] | = lim_{n →∞} | [(x^{n + 1})/((n + 1)²)]( [(n²)/(xⁿ)] ) | = lim_{n → ∞} | [(xn²)/(( n + 1 )²)] || x | < 1
Determine convergence
| x | < 1

− 1 < x < 1

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Answer

Power Series Convergence

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

Power Series

Definition

Radius & Interval of Convergence

Example 1

Example 2

Example 3

Example 4

Intro 0:00
Power Series 0:09

Definition

Radius & Interval of Convergence 2:07
Example 1 2:28
Example 2 4:18
Example 3 6:20
Example 4 9:11

Mathematics: AP Calculus BC

Section 1: Prerequisites
	Parametric Curves	10:10
	Polar Coordinates	14:54
	Vector Functions	12:05
Section 2: Differentiation
	Parametric Differentiation	13:15
	Polar Differentiation	10:14
	L'Hopital's Rule	6:43
Section 3: Integration
	Integration By Parts	19:04
	Integration By Partial Fractions	22:04
	Improper Integrals	19:01
Section 4: Applications of Integration
	Logistic Growth	19:16
	Arc Length for Parametric & Polar Curves	17:40
	Area for Parametric & Polar Curves	4:33
Section 5: Sequences, Series, & Approximations
	Definition & Convergence	7:10
	Geometric Series	10:59
	Harmonic & P Series	3:58
	Integral Test	11:31
	Comparison Test	7:47
	Limit Comparison Test	9:40
	Ratio Test	11:20
	Alternating Series	9:11
	Absolute Convergence	7:13
	Power Series Convergence	12:52
	Taylor Series	15:11
	Power Series Operations	16:40
	Lagrange Error	7:09
Section 6: Practice Test
	AP Calc BC Practice Test Section 1: Multi Choice Part 1	15:45
	AP Calc BC Practice Test Section 1: Multi Choice Part 2	21:38
	AP Calc BC Practice Test Section 1: Multi Choice Part 3	17:31
	AP Calc BC Practice Test Section 1: Multi Choice Part 4	14:17
	AP Calc BC Practice Test Section 1: Multi Choice Part 5	22:45
	AP Calc BC Practice Test Section 1: Free Response Part 1	9:55
	AP Calc BC Practice Test Section 1: Free Response Part 2	10:08
	AP Calc BC Practice Test Section 1: Free Response Part 3	13:58

Related Books

	Cracking the AP Calculus BC Exam, 2015 Edition Author: Princeton Review ISBN: 804124825 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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John Zhu

Power Series Convergence

Slide Duration:

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro