Does the Series Converge or Diverge for Different Values of z?

[erbilsilik]

Homework Statement

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

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

Homework Equations

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

The Attempt at a Solution

Using the ratio test I've found:

$$\lim _{p\rightarrow \infty }\sum _{p=1}^{\infty }\dfrac {z^{p}} {\left( p+1\right) ^{3/2}}$$
[/B]

 

[Fightfish]

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.