Q:

Simplify the following expression for small x

$$ \dfrac{\left(x+1\right)^2}{x^2\left(2x+1\right)}-\dfrac{\left(x+1\right)\ln\left(2x+1\right)}{x^3} $$
How to simplify the above expression for small $x$? I'm having trouble with $x^2$ and $x^3$ in the denominators. The other factors, I can Taylor expand but what to do with $x^2$ and $x^3$? The end result should be a simple expression like $a+bx+cx^2$. Hints are also fine :)

A:

Hint: For $|x| \ll 1$, we have
$$ \ln(1 + 2x) \approx 2x - 8x^3 + O(x^5) $$
$$ \frac{1}{1+2x} \approx 1 - 2x + 4x^2 + O(x^3) $$
Apply these approximations to your function. You'll still have a factor of $1/x^2$ in both terms. A simple polynomial approximation doesn't exist.

