Sign In | Subscribe
Start learning today, and be successful in your academic & professional career. Start Today!
Loading video...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Calculus BC
  • Discussion

  • Study Guides

  • Practice Questions

  • Download Lecture Slides

  • Table of Contents

  • Related Books

Bookmark and Share
Lecture Comments (1)

0 answers

Post by Wael Saleh on October 20, 2013

Great job, As you know there are many types of tests, Therefore, I have problem !!  Are there any certain signals in question describe which kind of tests should I use it to solve problem?
Best regards,

Definition & Convergence

  • Take the limit of the sequence to find its convergence:
    • If limit is finite, then sequence converges
    • If limit is infinite, then sequence diverges
  • Sequences are the building blocks for infinite series

Definition & Convergence

Does an = [3/(n2)] diverge?
Find the limit approaching ∞limn → ∞ [3/(n2)] = 00 is a rational number therefore an does not diverge
Does an = [( − 2)/(n + 1000)] converge?
Find the limit approaching ∞limn → ∞ [( − 2)/(n + 1000)] = 00 is a rational number therefore an does converge
Does an = [n/((n − 1)2)] converge or diverge?
  • Find the limit approaching ∞
  • limn → ∞ [n/((n − 1)2)] = limn → ∞ [n/(n2 − 2n + 1)]
  • Use the Rational Functions rule
  • limn → ∞ [n/(n2 − 2n + 1)] = [n/(n2)]
  • = [1/n]
  • = 0
0 is a rational number therefore an converges
Does an = [(n2 + 3n + 5)/(n + 1)] diverge?
  • Find the limit approaching ∞ using the Rational Functions rule
  • limn → ∞ [(n2 + 3n + 5)/(n + 1)] = [(n2)/n]
= ∞∞ is not finite so an diverges
Does an = [(18n3 + 5)/(9n3 + n2 + 9n + 1)] converge?
limn → ∞ [(18n3 + 5)/(9n3 + n2 + 9n + 1)] = [(18n3)/(9n3)] = 22 is a rational number therefore an does converge
Does an = [((2n − 3)2)/(n2)] converge or diverge?
  • Find the limit approaching ∞ using the Rational Functions rule
  • limn → ∞ [((2n − 3)2)/(n2)] = limn → ∞ [(4n2 − 6n + 9)/(n2)]
  • = [(4n2)/(n2)]
  • = 4
2 is a rational number therefore an does converge
Does an = [(n2 − 2n + 1)/(n + 1)] converge?
  • Find the limit approaching ∞
= limn → ∞ n − 1 = ∞∞ is not finite so an diverges
Does an = [(n4 − 2n3 + n2)/(3n3 + n + 96)] diverge?
  • Find the limit approaching ∞ using the Rational Functions rule
= [n/3] = ∞∞ is not finite so an diverges
Does an = ncosn converge or diverge?
Find the limit approaching ∞ using the Rational Functions rule
limn → ∞ ncosn = ∞∞ is not finite so an diverges
Does an = [sinn/(n2)] converge or diverge?
  • Find the limit approaching ∞ by considering the values of sine
  • Note that although sinn is always alternating values, n2 is always increasing as n apporaches ∞
  • limn → ∞ [sinn/(n2)] = 0
0 is a finite number, therefore an converges

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Definition & 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 0:00
  • Sequences: Definition 0:09
    • Definition
    • Example 1
  • Sequences: Convergence 2:02
  • Example 1 2:52
  • Example 2 3:36
  • Example 3 4:47
  • Example 4 6:16