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Testing series for convergence

  1. Apr 28, 2004 #1
    [tex]\sum_{n=1}^\infty\frac{\cos{\frac{n}{2}}}{n^2+4n}[/tex]


    Sorry I am just trying out this latex stuff as it's very new to me.

    Anyways, I want to test a series for convergence. The series is (if the latex does not work right) suppose to be

    The sum from n=1 to infinity [cos(n/2) / (n^2+4n)]

    Since this series has postive and negative terms but not alternating terms I have a limited number of test to try.

    I used the Ratio test.

    So I take lim as n goes to infinity of [cos((n+1)/2) / ((n+1)^2+4(n+1))] over
    [cos(n/2) / (n^2+4n)]

    and of course that is in absolute value bars.

    Now as n goes to infinity (n^2+4n)/((n+1)^2+4(n+1)) goes to one and I am left with the limit as n goes to inifinity of

    Absolute value[ cos((n+1)/2) / cos(n/2) ]

    Now I am sure this goes to one since my calculator can take this limit, but how could someone actually take this limit is another question. Can anyone help?

    Thanks

    P.S. I will work on this latex so maybe next time things will look better
     
    Last edited: Apr 29, 2004
  2. jcsd
  3. Apr 29, 2004 #2

    arildno

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    Hint:
    Sum(i from 1 to inf)1/(i^(2)) is convergent
     
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