Mathematics

Mathematics

Problem

The function g(x)=1+2x1−xg(x)=\dfrac{1+2x}{1-x} has a Maclaurin series that can be written as g(x)=∑n=0∞anxng(x)=\sum_{n=0}^{\infty}a_n x^n. Which of the following gives g(x)g(x) up to and including the x3x^{3} term? a) 1+2x+2x2+2x3+⋯1+2x+2x^{2}+2x^{3}+\cdots b) 1+x−x2+x3+⋯1+x-x^{2}+x^{3}+\cdots c) 1+3x+3x2+3x3+⋯1+3x+3x^{2}+3x^{3}+\cdots d) 1+3x+6x2+9x3+⋯1+3x+6x^{2}+9x^{3}+\cdots
Intuitive explanation

The key is to rewrite this rational function as a sum of simpler pieces you already know how to expand. Split the numerator strategically so you can use the geometric series 11−x=1+x+x2+x3+⋯\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, then multiply or add series together to capture all terms up to x3x^3. This turns an unfamiliar fraction into a familiar pattern.

Intuitive explanation

The key is to rewrite this rational function as a sum of simpler pieces you already know how to expand. Split the numerator strategically so you can use the geometric series 11−x=1+x+x2+x3+⋯\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, then multiply or add series together to capture all terms up to x3x^3. This turns an unfamiliar fraction into a familiar pattern.

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