Maxwell-Boltzmann Energy distribution

[WrongMan]

Homework Statement

find the average energy of a system with n energy states (0, 1E, 2E, 3E...nE)

Homework Equations

P(E) = e^-BE/Z - where B=1/KbT and Z= ∑e(^-BE)ⁿ
<E>=∑(nE* (e^-BE)ⁿ) /Z

The Attempt at a Solution

i feel like I've gone down the correct path - that is finding result of the sums.
Z - if S=R⁰+R¹+...+Rⁿ i do S-RS and get S=R⁰-Rⁿ⁺¹/1-R ... so Z= e⁰-e^-BE(n+1)/1-e^BE
now it gets tricky, i tried to evaluate the sum ∑(nE* (e^-BE)ⁿ) in a similar way,
so S₂=0R⁰+1R¹+2R²...
and RS₂=0R+1R²+2R³...
so S₂-RS₂=S-1 (remember S=Z; "-1" cause R⁰ is missng)
so S₂=(1-e^-BE(n+1)/(1-e^-BE)²-1/(1-e^-BE).

substituting into average fomula and simplifying my answer is: (1/1-e^-BE) - (1/1-e^BE(n+1))
is this correct?

 

[TSny]

now it gets tricky, i tried to evaluate the sum ∑(nE* (e^-BE)ⁿ) in a similar way,
so S₂=0R⁰+1R¹+2R²...
and RS₂=0R+1R²+2R³...
so S₂-RS₂=S-1 (remember S=Z; "-1" cause R⁰ is missng)

You're missing a term on the right hand side of the last equation. (Don't forget to consider the last terms in S₂ and RS₂).

 

[WrongMan]

You're missing a term on the right hand side of the last equation. (Don't forget to consider the last terms in S₂ and RS₂).

oh right RS₂ ends with nRⁿ⁺¹ so S₂ ends with "-nRⁿ⁺¹" (the missing term)
so my new answer is: (1/1-e^-BE) + (-1-ne^-BE(n+1)/1-e^-BE(n+1))

 

[TSny]

OK, that looks right for S₂ / Z. But that's not quite the answer for <E>.

When you have expressions like 1/1-e^-BE , you should include parentheses around the entire denominator: 1/(1-e^-BE).