# Bose function convergence

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1. May 21, 2016

### erbilsilik

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. May 21, 2016

### Fightfish

You applied the ratio test wrongly. Given a series
$$\sum_{p = 1}^{\infty} a_{p},$$
the ratio test involves looking at the quantity
$$\lim_{p \to \infty} \frac{a_{p+1}}{a_{p}}.$$

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

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