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Homework Help: Sequences converge or diverge?

  1. Mar 17, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to see if these sequences converge or diverge:

    1) [tex] a_n = ncosn\pi[/tex]

    2) [tex]{ 0,1,0,0,1,0,0,0,1,0,0,0,0,1,... }[/tex]

    3) [tex]a_n = \frac{1 . 3 . 5 . ... (2n - 1)}{n!}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    1) [tex]]cosn\pi = -1[/itex] so [itex]a_n \to -\infty[/tex]?

    2) What is this. where is the n?

    3) Not sure
  2. jcsd
  3. Mar 17, 2010 #2


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    1) are you sure cos(n*pi)=-1 always? What's cos(0) or cos(2*pi)?

    2) There is no n, it's a pattern sequence.

    3) What convergence/divergence tests have you thought of applying to this? Have you gotten anywhere with it? The last course I took that involved proving convergence/divergence was 3 years ago...so I don't remember every test that could be applied.
  4. Mar 17, 2010 #3


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    I wouldn't consider an->(-infinity) to be convergent. For 2) it appears to be a sequence with an infinite number of 0's and 1's in it. I wouldn't be distracted by trying to figure out a pattern. Who know's what in the '...'? It could be '0,0,0,0,0,0,0,0,0,0,0' forever. Just assume alternating 1's and 0's continue infinitely. What about convergence? For 3) you can apply a ratio test to a sequence. Each additional factor in the numerator is roughly twice the size of the corresponding factor the denominator. What's you gut feeling?
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