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I have seen the following two cases for convergence testing using Gauss's test-

$\frac{u_j}{u_{j+1}} = \frac{(2j+1)(2j+2)}{2j(2j+1)-\lambda}$

For large j, my textbook reformulates the RHS expression to-

$\frac{2j+2}{2j} + \frac{B(j)}{j^2} = 1 + \frac{1}{j} + \frac{b(j)}{j^2}$

In a different question, I need to reformulate the following expression -

$\frac{n^2 + a_{1}n + a_0}{n^2+b_1n+b_0}$

What is the right way to reformulate these expressions?

P.S: I do not have the reformulated expression for the second expression.

Paddy
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1 Answers1

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Just factor out the $n^2$ and use the usual $\dfrac1{1+x}=1-x+x^2+o(x^2)$ for $x\approx 0$. This is a technique you should master. So you get \begin{align*} \frac{n^2+a_1n+a_0}{n^2+b_1n+b_0} &= \frac{1+\frac{a_1}n+\frac{a_0}{n^2}}{1+\frac{b_1}n+\frac{b_0}{n^2}} \\ &= \big(1+\frac{a_1}n+\frac{a_0}{n^2}\big)\big(1-\frac{b_1}n+\frac{b_1^2-b_0}{n^2}+o(\tfrac1{n^2})\big). \end{align*} Can you finish? (You probably do not need this many terms, but you should learn the technique regardless.)

Ted Shifrin
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