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I am looking for an approximation of $\ln(1 + x)$ or $\ln(x)$.

We know $\ln(1 + x)$ has a Taylor expansion:

${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,}$

But for the problem I am working on, (a problem in random walk), I would like to get an alternative approximation of $\ln(1 + x)$, and hope it has some natural connections to combinatorics or probability or random walk.

Is there an approximation of $\ln(1 + x)$ or $\ln(x)$ which has some connections to combinatorics or probability or random walk ?

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At it's simplest, $C(x)=-\log(1-x)= \sum_{n > 0} (n-1)! x^n/n!$ is the exponential generating function (e.g.f.) for the Joyal species of cyclic permutations in the theory of combinatorial species. The number of ways of uniquely ordering the natural numbers up to $n$ on an oriented circle is $(n-1)!$.

Taking the log of functions is related to the Faber polynomials, and exponentiating the umbralized version of the log function is related in MO-Q "Cycling through the zeta garden: zeta functions for graphs, cycle index polynomials, and determinants" and its links to various areas of number theory, linear algebra, graph theory, operator calculus, dynamical systems, combinatorics, random walks, and more.

Also see an answer to this MO-Q.

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  • $\begingroup$ Hi, Tom, Thank you for the answer. I read the link you posted, but it is not clear how this will give an approximation of logarithm function with connection to combinatorics. Can you explain it a bit more ? Thank you. $\endgroup$
    – david
    Sep 5, 2019 at 0:34
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    $\begingroup$ It's a generalization, not approx, that pops up in advanced combinatorics, algebra, and topology. Simply replace each term of the Taylor series $x^k/k$ by $(a.x)^k/k = a_k x^k/k$ where $a_k$ is an arbitrary indeterminate. $\endgroup$ Sep 5, 2019 at 1:02
  • $\begingroup$ Hi, Tom: I am more interested in finding out a "nature connection" of log(1 + x) with random walk or combinatorics, not a generalization of log(1 + x). Are you aware something like this ? $\endgroup$
    – david
    Sep 5, 2019 at 1:21
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    $\begingroup$ The generaliztion has a very close connection to random walks on a graph. You need to dig some yourself. $\endgroup$ Sep 5, 2019 at 4:28

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