The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would be meaningful, but as this is a part of a longer proof, I think that you need to look at the context in order to understand the question. am thinking about a step in the proof of the equivalence of ensembles in StatMech on page 25. step from 3.19 to 3.20.
It seems to be argued there that for $A$ and $K$ large enough, the term
$$\sum_{N=0}^{\infty} \int_{-Bn}^{\infty} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $$
becomes arbitrarily small (We have $N = \frac{n}{|\lambda _m|}$ and $|\lambda_m| = \int_{\lambda_m}dx$), as the essential thing is to replace this whole term by $e^{\beta |\lambda_m|p(\beta, \mu)}$. $\beta, B$ are larger than zero.
Despite, I do not really trust this step cause if we assume that only $N>K$, then it seems to be necessary for this whole expression to converge that $B$ is in $(0,2)$. Thus, there seems to be an additional requirement regarding $B$ in this step. Does anybody see if there is an additional trick used here or am I just missing something?
In particular, I doubt that $$\sum_{N=0}^{\infty} \int_{-Bn}^{0} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $$ converges in general.
The thing is that if we assume $N>K$, then we have $$\sum_{N=K+1}^{\infty} \int_{-B \frac{N}{|\lambda_m|}}^{0} e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $$ and this expression converges even for large $K$ only if $B<2$, wouldn't you agree?- So I just don't understand this step. If you have any questions, please let me know.