If the expansion in powers of x of the function (1)/((1-a x)(1-b x)) is a0+a1 x+a2 x^2+a3 x^3+…,

step 1 in this problem we have 1 minus a x to the power of minus 1 into 1 minus b x to the power of min

If the expansion in powers of x of the function (1)/((1-a...

If the expansion in powers of $x$ of the function $\frac{1}{(1-a x)(1-b x)}$ is $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots$, then $\mathrm{a}_{\mathrm{n}}$ is (A) $\frac{b^{n}-a^{n}}{b-a}$ (B) $\frac{a^{n}-b^{n}}{b-a}$ (C) $\frac{a^{n+1}-b^{n+1}}{b-a}$ (D) $\frac{b^{n+1}-a^{n+1}}{b-a}$

If the expansion in powers of $x$ of the function $\frac{1}{(1-a x)(1-b x)}$ is $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots$, then $\mathrm{a}_{\mathrm{n}}$ is
(A) $\frac{b^{n}-a^{n}}{b-a}$
(B) $\frac{a^{n}-b^{n}}{b-a}$
(C) $\frac{a^{n+1}-b^{n+1}}{b-a}$
(D) $\frac{b^{n+1}-a^{n+1}}{b-a}$

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Key Concepts

Power Series Expansion

This concept involves writing a function as an infinite sum of powers of x, where each term’s coefficient reflects the function's behavior. It is a fundamental tool for approximating functions locally and analyzing their properties via their series representation.

Geometric Series

A geometric series, exemplified by the series 1/(1 - x) = 1 + x + x^2 + ... for |x| < 1, is a basic example of a power series. Understanding geometric series is key to expanding more complex rational functions by expressing them as sums of simpler series.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This decomposition simplifies the work of finding individual power series for each term, making it easier to derive the coefficients in the overall expansion.

Generating Functions

Generating functions encode sequences as coefficients in a power series. They are a powerful analytical tool in combinatorics and algebra, enabling the study and manipulation of sequences by working with their associated series representations.

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