Using division to find the power series representation
Replacing a polynomial with its power series expansion
Sometimes we’ll want to use polynomial long division to simplify a fraction, but either the numerator and/or denominator isn’t a polynomial.
In this case, we may be able to replace the non-polynomial with its power series expansion, which will be a polynomial.
The simplest way to do this for the non-polynomial is to find a similar, known power series expansion and then modify it to match the non-polynomial function.
Once we have polynomial expressions for both the numerator and denominator, we’ll do polynomial long division until we have the number of non-zero terms we’ve been asked for.
How to use division to find a power series expansion
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Turning an equation into a power series by using polynomial long division
Example
Use power series division to find the first three non-zero terms of the Maclaurin series of the given function.
???y=\frac{x}{e^{3x}}???
In order to use long division, we need polynomials in the numerator and denominator of our function. The numerator is already a polynomial, but we need to find a power series expansion for ???e^{3x}??? so that we can change it into a polynomial.
We know that the expanded version of the Maclaurin series for ???e^x??? is
???e^x=\sum^{\infty}_{n=0}\frac{1}{n!}x^n=1+x+\frac12x^2+\frac16x^3+...???
Since we have ???e^{3x}??? instead of ???e^x???, we’ll need to modify the series, letting ???x=3x???, such that the expanded series will be
???e^{3x}=1+3x+\frac12(3x)^2+\frac16(3x)^3+...???
???e^{3x}=1+3x+\frac92x^2+\frac92x^3+...???
Now that both the numerator and denominator are represented as polynomials, we’ll do the long division.
Remember, we only need to find the first three non-zero terms. We’ll take the first three terms from our quotient and say that the first three non-zero terms are
???y=\frac{x}{e^{3x}}\approx x-3x^2+\frac92x^3???
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