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Calculus BC Alternating Series

Section 5: Sequences, Series, & Approximations: Lecture 8 | 9:11 min

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Lecture Comments (3)

2 answers

Last reply by: Jimmy Trinh
Sun May 3, 2015 6:55 PM

Post by Angela Patrick on March 30, 2014

I believe in example 2, when he says hat the series diverges it's an incorrect statement because if the test fails it should be a no conclusion situation

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Alternating Series

Alternating negative and positive components
Alternating series converges if:

If given terms of a series, solve for proper series notation first ()

Alternating Series

Does the series ∑[(( − 1)^{n + 1})/(n + 1)] converge?

Check if a_{n + 1}< a_n
a_{n + 1} = [1/(n + 2)],a_n = [1/(n + 1)]
[1/(n + 2)] < [1/(n + 1)]
a_{n + 1}< a_n

Find lim_{n → ∞} a_nlim_{n → ∞} [1/(n + 1)] = 0 Thus the series converges

Does the series ∑[(2( − 1)^{n + 1})/(n − 3)] converge?

Check if a_{n + 1}< a_n
a_{n + 1} = [2/(n − 2)],a_n = [2/(n − 3)]
[2/(n − 2)] < [2/(n − 3)]
a_{n + 1}< a_n

Find lim_{n → ∞} a_nlim_{n → ∞} [2/(n − 3)] = 0 Thus the series converges

Does the series ∑[(( − 1)^{n + 1})/(n²)] converge?

Check if a_{n + 1}< a_n
a_n = [1/(n²)],a_{n + 1} = [1/((n + 1)²)]
[1/((n + 1)²)] < [1/(n²)]
a_{n + 1}< a_n

Find lim_{n → ∞} a_nlim_{n → ∞} [1/(n²)] = 0 Thus the series converges

Does the series ∑[(n³( − 1)^{n + 1})/(n²)] converge?

Check if a_{n + 1}< a_n
a_n = [(n³)/(n²)]
= n

n < [((n + 1)³)/((n + 1)²)]a_{n + 1}> a_n Thus the series diverges

Does the series [1/4] − [2/5] + [1/2] − [4/7] + ... + [(( − 1)^{n + 1}n)/(n + 3)] converge?

Determine a_{n + 1}
a_n = [n/(n + 3)],a_{n + 1} = [(n + 1)/(n + 4)]

a₅ = [5/8] = 0.625a₆ = [6/9] = 0.666...a₆> a₅, thus the series diverges

Does the series ∑[(( − 1)^{n + 1})/(3ⁿ)] converge?

Determine a_{n + 1}
a_n = [1/(3ⁿ)],a_{n + 1} = [1/(3^{n + 1})]
Try an arbitrary positive integer for n, such as n = 1
a₁ = [1/3] = 0.333...
a₂ = [1/9] = 0.111....
a₂< a₁

Find lim_{n → ∞} a_nlim_{n → ∞} [1/(3ⁿ)] = 0 Thus the series diverges

Does the series ∑[(( − 1)^{n + 1}n!)/(2ⁿ)] converge?

Determine a_{n + 1}
a_n = [n!/(2ⁿ)],a_{n + 1} = [(( n + 1 )!)/(2^{n + 1})]

a₂ = [2/4] = 0.5a₃ = [6/8] = 0.75a₃> a₂, thus the series diverges

Does the series ∑[(( − 1)^{n + 1}2ⁿ)/n!] converge?

Determine a_{n + 1}
a_n = [(2ⁿ)/n!],a_{n + 1} = [(2^{n + 1})/(( n + 1 )!)]
Try an arbitrary positive integer for n, such as n = 2
a₂ = [4/2]
a₃ = [8/6]
a₃< a₂

Find lim_{n → ∞} a_n using the Ratio Test
lim_{n → → ∞} [(2ⁿ)/n!] = 0 Thus the series converges

Does the series ∑( − 1)^{n + 1}2( [1/3] )^{n − 1} converge?

r = − [1/3]r < 1 Thus the series converges

What is the sum of the ∑( − 1)^{n + 1}2( [1/3] )^{n − 1} ?

Apply the Sum equation
Sum = [a/(1 − r)] = [2/(1 + [1/3])] = [2/([4/3])] = [6/4]

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Answer

Alternating Series

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

Alternating Series

Convergence Test

Example 1

Example 2

Example 3

Example 4

Intro 0:00
Alternating Series 0:08
Convergence Test 0:59
Example 1 1:27
Example 2 2:42
Example 3 4:57
Example 4 6:37

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

Alternating Series

Slide Duration:

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro