TITLE: Can you express $\ln(x)$ as a series with negative or fractional powers of $x$?
QUESTION [6 upvotes]: I read in my book that $\ln(x)$ cannot be written as a power series.  That is a series whose terms contain only non-negative integer powers of $x$.  But can $\ln(x)$ be written as an infinite series whose terms contain negative or fractional powers of $x$?  Then there is the bigger question.  Can all functions be expressed with such terms?

REPLY [4 votes]: The "problem" with Taylor series is, that they (have to!) converge in discs around the point of expansion. Laurent series have the same "problem" (rings around the point of expansion). In both cases the multi-value property of the ln function cannot be represented. To have any chance of convergence we have to restrict ourselves to a cut plan.
An example of a series with convergence in a cut plane is a series of Padé approximants (converges in complex plane without the negative real axis). And indeed, the Padé approximants converge to the main branch of the logarithm.
Such approximants consist of a polynomial devided by a polynomial. Eg
$$P_6^6(x) = \frac{7 \left(-7-132 x-375 x^2+375 x^4+132 x^5+7 x^6\right)}{10 \left(1+36 x+225 x^2+400 x^3+225 x^4+36 x^5+x^6\right)} .$$
This is the (6,6) Padé approximant around the expansion point 1. If you plot it, you will see, that it already resembles the logarithm farely well for small phases of the complex argument.
Note that the main Padés are equivalent to continued fractions. So if you don't like the term Padé approximant you might also say, that the logarithm can be represented as a continued fraction.