P4.4 - Binomial expansion

The binomial series expands a rational power around $x=0$. Put the expression into the standard form $(1+u)^n$ first; the coefficients come from the descending products of $n$, while the size of $u$ controls where a non-terminating series is valid.

Rewrite $(ax+b)^n=b^n\left(1+\frac{a}{b}x\right)^n$. For a negative or non-integer rational $n$, require $\left|\frac{a}{b}x\right|<1$, so $|x|<\left|b/a\right|$. A non-negative integer power terminates and does not need this convergence restriction.

For $(4+5x)^{1/2}$, take out $4^{1/2}=2$ and use $u=5x/4$: $$2\left(1+\frac12u-\frac18u^2+\frac1{16}u^3+\cdots\right)=2+\frac54x-\frac{25}{64}x^2+\frac{125}{512}x^3+\cdots.$$ This series is valid for $|5x/4|<1$, or $|x|<4/5$.

For a rational function, decompose first and expand each simple fraction. For example, $$\frac{2}{1-2x}+\frac{3}{4+x}=2(1-2x)^{-1}+\frac34(1+x/4)^{-1}$$ $$=\frac{11}{4}+\frac{61}{16}x+\frac{515}{64}x^2+\cdots.$$ The two component ranges are $|x|<1/2$ and $|x|<4$, so the combined expansion requires the narrower condition $|x|<1/2$.