Dividing power series (1): Long Division

Dividing power series is a a powerful method to obtain new series from known ones. In this thread, I will show some simple examples of dividing power series. We will start from the simplest example and move on the most intricate ones. The first example is that of a geometric series. The series for $\dfrac{1}{1-x}$ is worth learning by heart because it is ubiquitous in obtaining new power series. The scheme for dividing the series is represented below. This scheme is what we call long division of power series.

Series representation of $\dfrac{1}{1-x}$

Thus $\dfrac{1}{1-x}=1+x+x^2+x^3+x^4+x^5+x^6+...$

Series representation of $\dfrac{1}{1+x}$

Thus $\dfrac{1}{1+x}=1-x+x^2-x^3+x^4-x^5+x^6+...$

Series representation of $\dfrac{1}{1+2x}$

Thus $\dfrac{1}{1+2x}=1-2x+4x^2-8x^3+16x^4-32x^5+64x^6+...$

$=1-2x+2^2x^2-2^3x^3+2^4x^4-2^5x^5+2^6x^6+...$

Series representation of $\dfrac{1}{1+3x}$

Thus $\dfrac{1}{1+3x}=1-3x+9x^2-27x^3+81x^4-243x^5+729x^6+...$

$=1-3x+3^2x^2-3^3x^3+3^4x^4-3^5x^5+3^6x^6+...$

Series representation of $\dfrac{1}{1+4x}$

Thus $\dfrac{1}{1+4x}=1-4x+16x^2-64x^3+256x^4-1024x^5+4096x^6+...$

$=1-4x+4^2x^2-4^3x^3+4^4x^4-4^5x^5+4^6x^6+...$

Series of the form of $\dfrac{1}{1+kx}$

Here I will list the first 10 series in this form:

$\dfrac{1}{1+x}=1-x+x^2-x^3+x^4-x^5+x^6+...$

$\dfrac{1}{1+2x}=1-2x+2^2x^2-2^3x^3+2^4x^4-2^5x^5+2^6x^6+$

$\dfrac{1}{1+3x}=1-3x+3^2x^2-3^3x^3+3^4x^4-3^5x^5+3^6x^6+$

$\dfrac{1}{1+4x}=1-4x+4^2x^2-4^3x^3+4^4x^4-4^5x^5+4^6x^6+...$

$\dfrac{1}{1+5x}=1-5x+5^2x^2-5^3x^3+5^4x^4-5^5x^5+5^6x^6+...$

In general:

$\dfrac{1}{1+kx}=1-kx+k^2x^2-k^3x^3+k^4x^4-k^5x^5+k^6x^6+...$

We can insert fractional values to $k$ to obtain the inverse of these series:

$\dfrac{1}{1+\dfrac{1}{2}x}=\dfrac{2}{x+2}=1-\dfrac{x}{2}+\dfrac{x^2}{2^2}-\dfrac{x^3}{2^3}+\dfrac{x^4}{2^4}-\dfrac{x^5}{2^5}+\dfrac{x^6}{2^6}...$

$\dfrac{1}{1+\dfrac{1}{3}x}=\dfrac{3}{x+3}=1-\dfrac{x}{3}+\dfrac{x^2}{3^2}-\dfrac{x^3}{3^3}+\dfrac{x^4}{3^4}-\dfrac{x^5}{3^5}+\dfrac{x^6}{3^6}...$

$\dfrac{1}{1+\dfrac{1}{4}x}=\dfrac{4}{x+4}=1-\dfrac{x}{4}+\dfrac{x^2}{4^2}-\dfrac{x^3}{4^3}+\dfrac{x^4}{4^4}-\dfrac{x^5}{4^5}+\dfrac{x^6}{4^6}...$

$\dfrac{1}{1+\dfrac{1}{5}x}=\dfrac{5}{x+5}=1-\dfrac{x}{5}+\dfrac{x^2}{5^2}-\dfrac{x^3}{5^3}+\dfrac{x^4}{5^4}-\dfrac{x^5}{5^5}+\dfrac{x^6}{5^6}...$

Series of the form of $\dfrac{1}{1-kx}$

$\dfrac{1}{1-x}=1+x+x^2+x^3+x^4+x^5+x^6+...$

$\dfrac{1}{1-2x}=1+2x+2^2x^2+2^3x^3+2^4x^4+2^5x^5+2^6x^6+$

$\dfrac{1}{1-3x}=1+3x+3^2x^2+3^3x^3+3^4x^4+3^5x^5+3^6x^6+$

$\dfrac{1}{1-4x}=1+4x+4^2x^2+4^3x^3+4^4x^4+4^5x^5+4^6x^6+...$

$\dfrac{1}{1-5x}=1+5x+5^2x^2+5^3x^3+5^4x^4+5^5x^5+5^6x^6+...$

In general

$\dfrac{1}{1-kx}=1+kx+k^2x^2+k^3x^3+k^4x^4+k^5x^5+k^6x^6+...$