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Integral Test

  1. Oct 8, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]\sum_{n=1}^{\infty} \frac{n^{k-1}}{n^k+c}[/itex], where k is a positive integer.

    2. Relevant equations

    3. The attempt at a solution
    I found that it was discontinuous at [itex]x = (-c)^{1/k}[/itex]; and to determine if the sequence is decreasing, I took the

    derivative which is--I think--[itex]f'(x) = \frac{(k-1)x^{k-2}(x^k+c)-x^{k-1}(kx^{k-1}}{(x^k+c)^2}[/itex]
    I am not quite sure how to simplify this, nor am I certain on how to find the intervals which the sequence is decreasing.
  2. jcsd
  3. Oct 8, 2012 #2


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    Homework Helper

    Now that you've taken the derivative of f, ask yourself, what are the critical points? Those will allow you to find if f is decreasing as x → ∞.
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