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Converging/Diverging Series

  1. Jul 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Does the series ∑[n=1,∞) sin4n / 4^n converge or diverge?

    2. Relevant equations

    Ratio Test

    lim n->∞ | a_n+1 / a_n |

    3. The attempt at a solution

    By Ratio Test.

    Let a_n = sin(4n) / 4^n


    lim n->∞ | (sin (4n+1) / 4^n+1) / (sin 4n / 4^n) |

    Skipping a few steps..

    = | (sin(4n+1)/sin(4n)) * (4^n)/(4^n * 4^1) |

    = 1/4 * lim n->∞ (sin(4n+1)/sin(4n))

    Here's my problem. How do I take the limit of (sin(4n+1)/sin(4n))? Did I do the whole problem wrong? Should I have used Root test?
    Last edited: Jul 15, 2008
  2. jcsd
  3. Jul 15, 2008 #2


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    Science Advisor
    Homework Helper

    I don't think the limit works out, as both sines will keep on oscillating, it's just like an ordinary sine in that respect. I think you will find the root test more useful, as you suggested.
  4. Jul 15, 2008 #3
    Thank you, I will try that.
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