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Bose function convergence

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  1. May 21, 2016 #1
    1. The problem statement, all variables and given/known data

    How can I show that this series is convergent for z=1 and z<1 and divergent for z>1

    $$\sum _{p=1}^{\infty }\dfrac {z^{p}} {p^{3/2}}$$

    2. Relevant equations

    http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx

    3. The attempt at a solution

    Using the ratio test I've found:

    $$\lim _{p\rightarrow \infty }\sum _{p=1}^{\infty }\dfrac {z^{p}} {\left( p+1\right) ^{3/2}}$$
     
  2. jcsd
  3. May 21, 2016 #2
    You applied the ratio test wrongly. Given a series
    [tex]\sum_{p = 1}^{\infty} a_{p},[/tex]
    the ratio test involves looking at the quantity
    [tex]\lim_{p \to \infty} \frac{a_{p+1}}{a_{p}}.[/tex]

    If this quantity is greater than one, then the series diverges.
     
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