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Lecture Comments (1)

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Post by Melinda Berry on April 1, 2015

Hello, could you confirm that this is a geometric series? I thought that a geometric series had to have a common ratio. In this case, what is the common ratio? Thank you.

Harmonic & P Series

  • Harmonic series: , always diverges
  • P-series: , check P value for convergence

Harmonic & P Series

Does the series + [1/8] + [1/27] + [1/64] + ... converge?
  • Recognize series as a P - series
  • 1 + [1/8] + [1/27] + [1/64] + ... = ∑n = 1 [1/(n3)]
Analyse pp = 3 > 1, thus the series converges
Does the series 1 + [1/(3√{2})] + [1/(3√{3})] + [1/(3√{4})] + ... converge?
  • Recognize series as a P - series
  • 1 + [1/(3√{2})] + [1/(3√{3})] + [1/(3√{4})] + ... = ∑n = 1 [1/(n1/3)]
Analyse pp = [1/3] ≤ 1, thus the series diverges
Does the series 3 + [3/4] + [1/3] + [3/16] + ... diverge?
  • Recognize series as a P - series
  • 3 + [3/4] + [1/3] + [3/16] + ... = ∑n = 1 [3/(n2)]
  • = 3∑n = 1 [1/(n2)]
Analyse pp = 2 > 1, thus the series converges
Does the series ∑n = 1 [1/(n7/8)] converge or diverge?
Analyse pp = [7/8] ≤ 1, thus the series diverges
Does the series ∑n = 1 [1/(n[91/72])] converge or diverge?
Analyse p
p = [91/72]f1, thus the series converges
Does the series ∑n = 1 [1/(√{n3} )] converge or diverge?
  • Alter the series with exponent properties
  • n = 1 [1/(√{n3} )] = ∑n = 1 [1/(n[3/2])]
Analyse pp = [3/2] > 1, thus the series converges
Does the series ∑n = 1 [1/(6√{n4})] converge or diverge?
  • Alter the series with exponent properties
  • n = 1 [1/(6√{n4})] = ∑n = 1 [1/(n[4/6])]
Analyse pp = [4/6] ≤ 1, thus the series diverges
Does the series ∑n = 1 [(2n + 2)/(n2 + n)] converge or diverge?
  • Simplify series
  • n = 1 [(2n + 2)/(n2 + n)] = ∑n = 1 [(2(n + 1))/(n(n + 1))]
  • = ∑n = 1 [2/n]
  • = 2 ∑n = 1 [1/n]
Analyse pp = 1 ≤ 1, thus the series diverges
Does the series ∑n = 1 [(√n )/n] converge or diverge?
  • Alter the series with exponent properties
  • n = 1 [(√n )/n] = ∑n = 1 [1/(n[1/2])]
Analyse pp = [1/2] ≤ 1, thus the series diverges
Does the series ∑n = 1 [(√{n2 + 1} )/(√{n4 + n2} )] converge or diverge?
  • Alter the series with exponent properties
  • n = 1 [(√{n2 + 1} )/(√{n4 + n2} )] = ∑n = 1 [(√{n2 + 1} )/(√{n2(n2 + 1)} )]
  • = ∑n = 1 [(√{n2 + 1} )/(√{n2} √{n2 + 1} )]
  • = ∑n = 1 [1/n]
Analyse pp = 1 ≤ 1, thus the series diverges
Does the series 36 + 9 + 4 + [36/16] + ... converge or diverge?
  • Examine the series by having the denominator expressed as the powers of 2
  • 36 + 9 + 4 + [36/16] + ... = [36/(12)] + [36/(22)] + [36/(32)] + [36/(42)] + ...
  • = ∑n = 1 [36/(n2)]
  • = 36∑n = 1 [1/(n2)]
Analyse p
p = 2 > 1,thus the series 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

Harmonic & P 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 0:00
  • Harmonic Series 0:08
  • P-Series 1:17
  • Example 1: P-Series Test 2:22
  • Example 2: P-Series Test 3:06