Express $\frac x{x^2-3x + 2}$ in the partial fraction form and show that its partial fraction $x^3$ can be neglected

Question

I am having trouble solving a multi part question.

Express $ \frac x{x^2-3x + 2} $ in the partial fraction form.

The answer I got was $\frac2{x-2}-\frac1{x-1}$ .

The problem comes when they asked:

Show that, if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then:

$$\frac x{x^2-3x +2}\approx\frac12x+\frac34x^2$$

I understand that I had to expand the partial fraction using Generalized Binomial Theorem such that I needed to expand: $-\left[1+\left(-\frac12x\right)\right]^{-1}+\left[1+\left(-x\right)\right]^{-1}$ since I needed to manipulate the equation into $\left(1+a\right)^n$ format.

I got $$\frac12x+\frac34x^2+\frac78x^3+\frac{15}{16}x^4+\cdots$$

but I am not sure how to continue to show that , if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected. Do i just sub in numbers ranging from -1 to 1 and that's it or is there a structured way to show?

Answer 1

I believe you are already done and just have linguistic issue; I think that "if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then [...]" should be interpreted as "if $x$ has property $P(x)$, then [...]" where $P(x)\equiv$ "$x^3$ and higher powers can be neglected". That is, you are not supposed to show that you can neglect $x^3$, it's an assumption. Thus, it's the same question as "show that power series expansion is $x/2 + 3x^2/4\, + $ higher order terms."

Answer 2

Just cross out terms with $x^3$, $x^4$, ... in the approximation.

The question assumes these terms are too small, and we don't care.

For example, if I know x is between 0 and 0.1. Then $x^3$ is at most 0.001. $x^4$ is even smaller.

I think the sum of the terms $x^3$, $x^4$, ... are at least smaller than $x^2$, which is at most 0.01.

I forget the exact formula for the errors. I just use sum of a geometric series and approximate the coefficient of all terms as 1.

Let's say an error of 0.02 is acceptable in the approximation.

I would use just the $x$ and $x^2$ terms in that case.

Less terms means a computer program can approximate the thing faster.