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Use ratio test to find radius and interval of convergence of power series

  1. Oct 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Use the ratio test to find the radius of convergence and the interval of convergence of the power series:

    [[Shown in attachment]]

    2. Relevant equations

    an+1/an=k

    Radius of convergence = 1/k

    Interval of convergence: | x-a |∠ R

    3. The attempt at a solution

    I began by finding the summation which I concluded was:

    Ʃ (2^n)(x-0)^n / n(factorial)

    So an+1/an = [2n+1/(n+1)(factorial)] *times* [ n(factorial)/ 2n ]

    After cancelling, I arrived at 2/(n+1)

    If that lim n→∞ is taken, it would be 0, meaning the radius of convergence is ∞, and the interval is from -∞ to ∞. Is that true or did I make a mistake, because the website for my homework won't allow me to enter infinity (∞)?
     

    Attached Files:

  2. jcsd
  3. Oct 30, 2012 #2

    Mark44

    Staff: Mentor

    What happened to x?

    BTW, the symbol for factorial is !, so you can write n! instead of n factorial.
     
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