I have two quantities $a(t)$ and $b(t)$ that have a constant mean ($a$ and $b$) and some small fluctuating noise part with vanishing mean $\delta a(t)$ and $\delta b(t)$. I'll write them as $a(t) = a + \delta a(t)$ and the same for $b(t)$. I also have some final quantity that is linearly proportional to $a(t)/b(t)$, and I want to find the noise spectral density of this quantity which will depend on the two fluctuating delta terms.
Now, I already have the noise spectral densities of $\delta a(t)$ and $\delta b(t)$ to start with. So I figured I'd write down $\frac{a+\delta a(t)}{b + \delta b(t)}$ and try to obtain some expression (up to some order) in terms of constant parts and time fluctuating parts, which will then give me an expression for the spectral density.
My problem is however that I do not know how to approximate this fraction for small delta terms. Of course the first step I thought of is splitting them up, so that we have $\frac{a}{b + \delta b(t)}$ (which I can then taylor expand) and $\frac{\delta a(t)}{b + \delta b(t)}$. This second term however, is giving me some trouble. What would be a valid approximation for this? If I simply say $b > \delta b(t)$ and drop the delta term I do have an expression, but this feels a bit shaky.
If $|\delta b| < |b|$, you have a geometric series expansion $$ \frac{1}{b + \delta b} = \frac{1}{b(1 + \frac{\delta b}{b})} = \frac{1}{b} \left[1 - \frac{\delta b}{b} + \left(\frac{\delta b}{b}\right)^{2} - \dots\right] = \frac{1}{b} \sum_{k=0}^{\infty} \left(-\frac{\delta b}{b}\right)^{k}. $$