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I wanted to produce some plots of the action of the Gauss shift map on cumulative distribution functions. This means I wanted to plot functions $F_n(x)$, for $0 \leq x \leq 1$, defined by $F_1(x) = x$ and $$F_{n+1}(x) = \sum_{k=1}^{\infty} \left( F_{n}\left( \tfrac{1}{k} \right) - F_n\left( \tfrac{1}{k+x} \right) \right).$$ The $F_n(x)$'s are cumulative distribution functions, meaning they increase monotonically from $F_n(0)=0$ to $F_n(1)=1$.

Here is my first attempt:

f[n_, x_] := If[n == 1, x, Sum[f[n - 1, 1/k] - f[n - 1, 1/(k + x)], {k, 1, Infinity}]]

This computes f[2,x] just fine and even gives a closed form EulerGamma + PolyGamma[0, 1 + x]. But trying Plot[f[3, x], {x, 0, 1}] gives the error "Sum: Sum does not converge".

Here is my second attempt:

g[n_, x_] := If[n == 1, x, Sum[g[n - 1, 1/k] - g[n - 1, 1/(k + x)], {k, 1, 20}]]

Now the functions plot fine, but it is visibly obvious that g[2, 1] and g[3, 1] are significantly less than $1$, being about $0.95$ and $0.88$ respectively. If I try raising the bound 20 high enough to solve this, the sum takes too long to compute.

Here is my third attempt:

h[n_, x_] := If[n == 1, x, NSum[h[n - 1, 1/k] - h[n - 1, 1/(k + x)], {k, 1, Infinity}]]

Now Plot[h[3, x], {x, 0, 1}] produces an empty plot. Trying Plot[Evaluate[h[3, x]], {x, 0, 1}] runs forever without returning. I also tried the f[n_, x_] := f[n, x] = trick in all of these variants without success.

What I think I want is a version of NSum which is intelligent enough to only go as deep in the sum as necessary to make the plot. Is there a way to do this, or some other good way to approach the problem?

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  • $\begingroup$ The two definitions only differ in that the LaTeX writes $F_{n+1}$ as a sum of $F_n$'s and the Mathematica writes f[n] as a sum of f[n-1]. Since n is a dummy variable, they are equivalent. $\endgroup$ Dec 5, 2019 at 20:42
  • $\begingroup$ If you'll accept a link of questionable legality, see page 362, equation (23) of doc.lagout.org/science/0_Computer%20Science/2_Algorithms/… $\endgroup$ Dec 5, 2019 at 20:44
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    $\begingroup$ Indeed your Latex definition doesn't agree with your code. Please check your parenthesis. $\endgroup$
    – asterix314
    Dec 6, 2019 at 5:50
  • $\begingroup$ Oh, it's the LaTeX that's buggy! Thanks. $\endgroup$ Dec 6, 2019 at 14:34
  • $\begingroup$ @DES Thank you for fixing the formula. The link to the definition you provided in comments unfortunately doesn’t work for me (it gets stuck tying to load and never shows anything). $\endgroup$
    – MarcoB
    Dec 7, 2019 at 14:44

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