Average energy for n-state system

[danyull]

Homework Statement

Find the average energy $\langle E \rangle$ for
(a) an n-state system in which a given state can have energy 0, ε, 2ε, 3ε... nε.
(b) a harmonic oscillator, in which a state can have energy 0, ε, 2ε, 3ε... (i.e. with no upper limit).

Homework Equations

Definition of temperature: $β = \frac 1 {K_BT} = \frac {d lnΩ(E)} {dE}$
Boltzmann distribution: $P(ε) ∝ e^{-εβ}$

The Attempt at a Solution

Since the energy here takes on discrete values, the average is found by taking the weighted sum of the probabilities, $$\langle E \rangle = \sum_{n=0}^n nε~e^{-nεβ}$$
and in the case of part (b), the sum goes to infinity. My problem is I don't know how to evaluate these sums. Any help would be appreciated, thanks!

 

[Fightfish]

One commonly used trick to evaluate such sums is the following observation that
$$n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } (e^{-n\varepsilon \beta})$$
and use the fact that the partial derivative commutes with the summation to get
$$\sum_{n=0}^{N} n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } \sum_{n=0}^{N} e^{-n\varepsilon \beta}$$
The summation is now simply just a geometric series.
(and don't forget to normalize your expectation value)