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Lecture Comments (1)

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Post by Milan Ray on May 3, 2014

What if the comparisons don't work? Then what to do?

Comparison Test

  • Compares series with known convergence properties with unknown
  • Comparison test of convergence:
  • If denominator term increases while numerator remains constant, then the entire term decreases

Comparison Test

Does the series ∑ [1/(2 + n4)] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(n4)]
  • Consider n = 2, then an ≤ un
[1/(n4)] p = 4 > 1, thus the series un converges.
Since un converges, then an converges
Does the series ∑ [1/(n2 − 50)] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(n2)]
  • Consider n = 3, then an ≤ un
[1/(n2)] p = 2 > 1, thus the series un converges
Since un converges, then an converges
Does the series ∑ [1/(√{n + 6})] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(n1/2)]
  • Consider n = 3, then anun
  • Determine ∑ un diverges with P - series properties
  • [1/(n1/2)]
  • p = [1/2] ≤ 1, thus the series un diverges
Since un diverges, then an diverges
Does the series 1 + [1/2] + [1/(√7 )] + ... + [1/(√{3n − 2} )] + ... converge?
  • Represent the series as a summation
  • 1 + [1/2] + [1/(√7 )] + ... + [1/(√{3n − 2} )] + ... = ∑ [1/(√{3n − 2} )]
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(√{3n} )] = [1/(√3 )]( [1/(√n )] )
  • Consider n = 2, then anun
[1/(√{3n} )] = ∑ [1/(√3 )]( [1/(√n )] ) = [1/(√3 )]∑ [1/(√n )] p = [1/2] ≤ 1, thus the series un diverges
Since un diverges, then an diverges
If ∑ un diverges and an ≤ un, does ∑ an diverge?
Analyze the conditions for the Comparison Test
No, because the condition requires anun
Does the series ∑ [(n + 1)/(n2 + 2)] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(n)]
  • Consider n = 3, then anun
[1/(n)] Since it's a Harmonic Series, the series un diverges
Since un diverges, then an diverges
Does the series ∑ [1/(√{n3 + 6} )] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(√{n3} )]
  • Consider n = 3, then an ≤ un
  • Determine ∑ un converges with P − series properties
  • [1/(√{n3} )] = ∑ [1/(n3/2)]
  • p = [3/2] > 1, thus the series un converges
Since un converges, then an converges
If ∑ un converges and anun, does ∑ an converge?
Analyze the conditions for the Comparison Test
No, because the condition requires an ≤ un
Does the series ∑ [(n3)/((n + 1)2)] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = n
  • Consider n = 3, then an ≤ un
n = ∞ Thus the series un diverges
Since un diverges, then an diverges
Does the series ∑ [1/(5n3 + n2 + 1)] converge?
  • Determine an appropriate un for the Comparison Test
  • Consider un = [1/(n3)]
  • Consider n = 2, then an ≤ un
  • Determine ∑ un converges with P - series properties
  • [1/(n3)]
  • p = 3 > 1, thus the series un converges
Since un converges, then 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

Comparison Test

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
  • Comparison Test 0:10
  • Example 1 1:07
  • Example 2 2:33
  • Example 3 4:20
  • Example 4 6:29