Why does this equality hold?
$1-\left(1+\frac{\lambda}{m}\right)^{-1} \left(1+\frac{\lambda}{m-1} \right)^{-1} \ \dots \ \left( 1+\frac{\lambda}{m-k+1} \right)^{-1}=\left(\frac{1}{m}+\frac{1}{m-1}+ \dots + \frac{1}{m-k+1} \right)\lambda + O(\lambda^2)$
as $\lambda \rightarrow 0$
Just found that on a paper with no additional information. I'm new to the Big-Oh-Notation and have absolutely no clue how to get from the left hand side to the right. Why is there a $O(\lambda^2)$ as part of a sum? Oh, and I'd be hella grateful if someone could explain the algebraic steps from left to right as well.
$$\left(1+\frac\lambda n\right)^{-1}=1-\frac\lambda n+\frac{\lambda^2}{n^2}\cdots$$
When you multiply all the factors and subtract from one, what remains is an entire series in $\lambda$ with no constant term. The linear term is as given and the next terms are of order $\lambda^2$ and more. This is just a Taylor development to the first order, and a remainder.