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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!

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    $\begingroup$ 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. $\endgroup$ – user14717 Dec 13 '18 at 19:37
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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 $10^{-300}$.

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