Why do harmonic series converge in a finite precision number system?

I'm new to numerical analysis. I'm still unclear as to why the harmonic series $$\sum 1/k$$, where k = 1, ... , infinity converges. I would appreciate any help! I can show that the sum does indeed converge in Matlab, I'm just unclear as to the reasoning behind it.

Thanks!

• In finite precision, there is a smallest representable number. At some point, 1/k is less than this smallest number, and must be rounded to zero. All successive terms are then zero. – user14717 Dec 13 '18 at 19:37

Simple situation: fixed point precision. Then there is a smallest number in the system, and once $$1/n$$ is less than that number, the "numerical harmonic series" no longer changes its value.
Slightly more complicated situation: floating point precision. Then there is a largest representable number $$\epsilon_{mach}$$ such that the numerical result of $$1+\epsilon_{mach}$$ is just $$1$$ (but the same result doesn't hold for $$0.1+\epsilon_{mach}$$). Once $$1/n<\epsilon_{mach}$$, certainly the harmonic series up to $$n-1$$ will exceed $$1$$ (since the first term is already $$1$$), so again the series will no longer change. Note that this happens long before the smallest representable number in the system is reached; for example in IEEE double precision, $$\epsilon_{mach}$$ is around $$10^{-16}$$ while the smallest representable number is around $$10^{-300}$$.