Simplify expression assuming b>>c using big O notation

Question

So I found in Approximating for $a \gg b$

that I can use big O notation to emulate a>>b behavior. However, when I try that with the following equation, I get an undesired result.

Expression:

Normal[((c + 2 b) a)/((c + d) (c + b)) /. {b -> b + O[c]}]

Expected:

2a/(c+d)

Actual return:

2a/d

Where in the above, I'm trying to state that b>>c. Therefore the b in the numerator and the b in the denominator should just cancel.

Is there something I'm missing?

Answer 1

I think you can just do a series expansion of b at infinity. For example:

Series[((c+2 b) a)/((c+d) (c+b)), {b, Infinity, 0}] //TeXForm

$\frac{2 a}{c+d}+O\left(\left(\frac{1}{b}\right)^1\right)$

For your example in the comments:

Series[((c+2 b) a)/((c+d) (c+b)(b+e)), {b, Infinity, 1}] //TeXForm

$\frac{2 a}{b (c+d)}+O\left(\left(\frac{1}{b}\right)^2\right)$

If only c is much smaller than b, you can use Daniel's answer in the comments:

Normal[Series[((c+2 b) a)/((c+d) (c+b)(b+\[Epsilon])) /. b -> c/\[Epsilon], {\[Epsilon], 0, 2}]] /. \[Epsilon]->c/b

(2 a)/(b (c + d)) - (a c)/(b^2 (c + d))

Compare this to the above approach where everything is smaller than b:

Normal @ Series[((c+2 b) a)/((c+d) (c+b)(b+e)), {b, Infinity, 2}]

(2 a)/(b (c + d)) + (-a c - 2 a e)/(b^2 (c + d))

Answer 2

Big $O$ notation is used to write a term to first order in a selected term. Here starting with the term

$$\frac{a\left(2b+c\right)}{\left(b+c\right)\left(c+d\right)}$$

and assuming that $b\gg c$, the goal is to write the fraction in first order in $\frac{b}{c}$. By multiplying the numerator and denominator by $1/b^2$ and making the substitutions $\tilde{a}\equiv a/b$, $\tilde{c}\equiv c/b$, and $\tilde{d}\equiv a/d$ we can rewrite the original term as

$$\frac{\tilde{a}\left(2+\tilde{c}\right)}{\left(1+\tilde{c}\right)\left(\tilde{c}+\tilde{d}\right)}$$

Now assuming that $b\gg c$ results in $\tilde{c}\approx 0$, thus we can do a Taylor expansion about the point $c=0$ using Series.

Series[(a (2 + c))/((1 + c) (c + d)), {c, 0, 0}]

this will give the output

2a/d + O[c]^1

Simplifying the original equation when $b\gg c$ is not as simple as just crossing out the $c$ terms in the numerator and the denominator due to the presence of the additional $c$ dependent cross terms resulting from the multiplication in the denominator.