What is the mistake in converting this sum into an integral? Why is the integral negative?

Question

Suppose $0<B<A$ and I am looking at the summation $$ S(N) = \sum_{k=0}^{N-1} \frac{\Delta x}{A - k\Delta x} $$ where $A - N\Delta x = B$. The summand is of the form $\Delta x / x_{k}$ where $x_{0} = A$, $x_{1} = A-\Delta x$, $x_{2} = A-2\Delta x$, etc. Now when I send $N\rightarrow\infty$, I expect to get a Riemann integral out, so I expect $$ \lim_{N\rightarrow \infty} S(N) = \int_{A}^{B} \frac{dx}{x}. $$ However, when I make the evaluation, I get $$ \int_{A}^{B} \frac{dx}{x} = \ln B - \ln A = \ln \left(\frac{B}{A}\right) < 0. $$ This is a negative number, because $B<A$. However, the summation is positive, so my simple question is, where did I go wrong?

Answer 1

It's because you've miscalculated $\Delta x.$ To obtain it, we take "upper" limit (in this case, $B$) minus "lower" limit (in this case, $A$), divided by $N.$ That means that $\Delta x$ is necessarily negative, and that $$B=A+N\Delta X.$$ Also, for any $0\leq k<N,$ we have $0<B<x_k:=A+k\Delta x,$ and so your summands $\frac{\Delta x}{x_k}$ are all negative.

The potentially confusing terminology is "upper" limit and "lower" limit. By this, we don't necessarily mean the greater of the two limits and the lesser. Rather, we indicate where those limits are written next to the integral sign (the upper end or lower end).