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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

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 an + 1 < an
  • an + 1 = [1/(n + 2)],an = [1/(n + 1)]
  • [1/(n + 2)] < [1/(n + 1)]
  • an + 1 < an
Find limn → ∞ anlimn → ∞ [1/(n + 1)] = 0 Thus the series converges
Does the series ∑[(2( − 1)n + 1)/(n − 3)] converge?
  • Check if an + 1 < an
  • an + 1 = [2/(n − 2)],an = [2/(n − 3)]
  • [2/(n − 2)] < [2/(n − 3)]
  • an + 1 < an
Find limn → ∞ anlimn → ∞ [2/(n − 3)] = 0 Thus the series converges
Does the series ∑[(( − 1)n + 1)/(n2)] converge?
  • Check if an + 1 < an
  • an = [1/(n2)],an + 1 = [1/((n + 1)2)]
  • [1/((n + 1)2)] < [1/(n2)]
  • an + 1 < an
Find limn → ∞ anlimn → ∞ [1/(n2)] = 0 Thus the series converges
Does the series ∑[(n3( − 1)n + 1)/(n2)] converge?
  • Check if an + 1 < an
  • an = [(n3)/(n2)]
  • = n
n < [((n + 1)3)/((n + 1)2)]an + 1 > an Thus the series diverges
Does the series [1/4] − [2/5] + [1/2] − [4/7] + ... + [(( − 1)n + 1n)/(n + 3)] converge?
  • Determine an + 1
  • an = [n/(n + 3)],an + 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 an + 1
  • an = [1/(3n)],an + 1 = [1/(3n + 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 limn → ∞ anlimn → ∞ [1/(3n)] = 0 Thus the series diverges
Does the series ∑[(( − 1)n + 1n!)/(2n)] converge?
  • Determine an + 1
  • an = [n!/(2n)],an + 1 = [(( n + 1 )!)/(2n + 1)]
a2 = [2/4] = 0.5a3 = [6/8] = 0.75a3 > a2, thus the series diverges
Does the series ∑[(( − 1)n + 12n)/n!] converge?
  • Determine an + 1
  • an = [(2n)/n!],an + 1 = [(2n + 1)/(( n + 1 )!)]
  • Try an arbitrary positive integer for n, such as n = 2
  • a2 = [4/2]
  • a3 = [8/6]
  • a3 < a2
Find limn → ∞ an using the Ratio Test
limn → → ∞ [(2n)/n!] = 0 Thus the series converges
Does the series ∑( − 1)n + 12( [1/3] )n − 1 converge?
r = − [1/3]r < 1 Thus the series converges
What is the sum of the ∑( − 1)n + 12( [1/3] )n − 1 ?
Apply the Sum equation
Sum = [a/(1 − r)] = [2/(1 + [1/3])] = [2/([4/3])] = [6/4]

*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

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 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