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How do we go from

\begin{equation*} \frac{1}{1+(\frac{x}{2})+(\frac{x^2}{3})+(\frac{x^3}{4})+\dots} \end{equation*}

to

\begin{equation*} 1-\left(\frac{x}{2}\right)-\left(\frac{x^2}{12}\right)-\dots? \end{equation*}

If I could consolidate the series in denominator in some form, then I could use binomial expansion in numerator by raising it to power $-1$, but I don't see the one in denominator in any known form other than converting it to $\log.$

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    @Moron, @Mariano: I was both trying to clarify as the question now reads and asserting some more of the terms of the result (as generated using Mathematica) because the first three terms didn't give me the slightest realistic clue what the pattern might be.2010-09-26

2 Answers 2

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Using $-\log(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \dots$

We have that

$ - \frac{x}{\log(1-x)} = \frac{1}{1 + \frac{x}{2} + \frac{x^2}{3} + \dots}$

Now the series for $\frac{x}{\log(1-x)}$ is well known.

See the question asked on this very site here: Formula for the harmonic series $H_n = \sum_{k=1}^n 1/k$ due to Gregorio Fontana

And the page here: http://en.wikipedia.org/wiki/Euler-Mascheroni_constant. (search the page for Gregory).

The series expansion is given by

$\frac{x}{\log(1-x)} = \sum_{k=0}^{\infty} C_{k} x^{k} = -1 + \frac{x}{2} + \frac{x^2}{12} + \frac{x^3}{24} + \dots$

The $C_{k}$ are called as Gregory coefficients. The wiki page I linked above tells you how they can be calculated using a recursive formula.

So to answer your question, we get

$\frac{1}{1 + \frac{x}{2} + \frac{x^2}{3} + \dots} = - \sum_{k=0}^{\infty} C_{k} x^k = 1 - \frac{x}{2} - \frac{x^2}{12} - \frac{x^3}{24} - \dots$

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    @Harpreet: Use the dollar signs and latex between them. Example `$x^2+y^2=z^2$` looks like $x^2+y^2=z^2$2010-09-25
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Let $f(x) = \sum f_n x^n$ be a formal power series. To compute the inverse of $1 - x f(x)$, write

$\frac{1}{1 - x f(x)} = \sum_{n \ge 0} x^n f^n(x).$

To compute the coefficient of $x^k$ it suffices to compute the contributions from the first $k$ terms on the RHS, so this results in a finite algorithm which will let you compute any particular coefficient of the inverse. If $f$ has special properties (for example it is a polynomial or more generally meromorphic) then often one can even find a nice closed form.