Changing Summation to Integral This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However, I don't see how. Can anyone justify this change?

 A: I think Reif was being sloppy there. I don't have the book handy but that seems to be the case. A passage from discrete to continuous would go something like this:
Say we're talking about the normalization of a probability mass function:
$$ 1= \sum_{n=0}^N W(n) \,. $$
You first note that $n$ is a counting number, so its increments are by 1. Namely, ${\rm d}n = 1$. So you can just insert one:
$$ 1 = \sum_{n=0}^N {\rm d}n W(n) \,.$$ 
Then for some reason, you might be interested in the variable $r=n/N$. The ratio of rightward steps or something. You rewrite:
$$ 1 = \sum_{n=0}^N \frac{{\rm d}n} {N} N W(n) \,.$$
Now you realize that as $N\to\infty$, the quantity ${\rm d}n/N$ approaches an  infinitesimal ${\rm d}r$ . Now, thanks to the infinitesimal element, the summation looks like a Riemann sum, allowing the passage to an integral
$$ 1= \sum_{n=0}^N \frac{{\rm d}n} {N} N W(n) \to \int_0^\infty {\rm d} r \, N W(Nr) =1 \,.$$
Then, one proclaims that a new function, say, $P(r)$ has to be the probability density for the continuous variable $r=n/N$ such that $$P(r) = N W(N r) \,.$$
From what I can see, Reif's treatment is not emphasising that the new variable is proportional to $1/N$, and also failing to identify that the probability density of the new variable is not the same $(W)$ as that of the old variable, even though they are very closely connected (equation above).
