Bose function convergence

If this quantity is less than one, then the series converges. If this quantity is exactly one, then the test is inconclusive.In summary, the conversation discusses how to show the convergence and divergence of a given series for different values of z. The ratio test is suggested as a method to determine the convergence or divergence, but it is applied incorrectly.
  • #1
erbilsilik
20
2

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]
 
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  • #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.
 

What is Bose function convergence?

Bose function convergence is a mathematical concept used to describe the behavior of a Bose-Einstein gas at low temperatures. It refers to the phenomenon in which the number of particles in the ground state of the system increases as the temperature decreases, ultimately leading to a sharp peak in the Bose-Einstein distribution function.

How is Bose function convergence calculated?

The Bose-Einstein distribution function, which describes the average number of particles occupying a given energy state, can be calculated using the Bose-Einstein distribution equation. This equation takes into account the temperature, energy of the state, and the chemical potential of the system.

What is the significance of Bose function convergence?

Bose function convergence is important in understanding the behavior of Bose-Einstein condensates, which are a state of matter formed at extremely low temperatures. It also has applications in quantum mechanics and statistical mechanics.

What factors can affect Bose function convergence?

The temperature, energy levels of the system, and the number of particles present can all influence Bose function convergence. Additionally, the type of particles (bosons or fermions) and the presence of external forces can also play a role.

Are there any real-world applications of Bose function convergence?

Yes, Bose function convergence has practical applications in fields such as quantum computing, atomic and molecular physics, and ultra-cold atomic systems. It is also being studied in the context of superfluidity and quantum phase transitions.

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