If the expansion in powers of x of the function 1(1-ax)(1-bx) is a_0+a_1x+a_2x^2+a_3x^3+, …

Question

If the expansion in powers of $x$ of the function $\dfrac{1}{(1-ax)(1-bx)}$ is $a_0+a_1x+a_2x^2+a_3x^3+\cdots$, then $a_n$ is:

Accepted Answer

$(1-ax)^{-1}(1-bx)^{-1}=(1+ax+a^2x^2+\cdots)(1+bx+b^2x^2+\cdots)$.

Coefficient of $x^n$: $b^n+ab^{n-1}+a^2b^{n-2}+\cdots+a^{n-1}b+a^n$ --- a G.P. with ratio $a/b$, first term $b^n$, $n+1$ terms:

$$a_n=\frac{b^n\left[1-(a/b)^{n+1}\right]}{1-a/b}=\frac{b^{n+1}-a^{n+1}}{b-a}.$$

Solutions · 1