Approximation of product of numbers close to 1 In one of the papers I came across to the approximation below:
$$\prod_{k=1}^K (1-x_k) \approx 1 - \sum_{k=1}^K x_k,$$
where $x_k$s are very small positive numbers.
I am aware that by the binomial approximation, we can write that
$$(1-x)^K \approx 1-Kx$$
, where $|x|<1$. I assume that these two approximations are related to each other but I am not able to solid connection between them.
Can you explain how to calculate the first approximation and when it is valid?

Use it for $K=3$
$$\prod_{k=1}^3 (1-x_k)=1- (x_1+x_2+x_3)+(x_1x_2+x_1x_3+x_2x_3)-x_1x_2x_3$$
You could do it in another way
$$P_K=\prod_{k=1}^K (1-x_k)\implies \log(P_K)=\sum_{k=1}^K \log(1-x_k)$$ Now, by Taylor
$$\log(1-x_k)=-x_k-\frac{x_k^2}{2}+O\left(x_k^3\right)$$
$$\log(P_K)=-\sum_{k=1}^K x_k-\frac 12\sum_{k=1}^K x^2_k$$
$$P_k=e^{\log(P_K)}=1-\sum_{k=1}^K x_k-\cdots$$
