How to find the maximum of this equation $x(1 - (1 - \frac{1}{x})^K)$ For x in reals and K some positive integer, can you find the maximum of this function analytically?
$f(x) = x(1 - (1 - \frac{1}{x})^K)$

When $K = 1$, $f(x) = 1$. So assume $K > 1$ (not necessarily an integer).
The derivative of your function is
$$ 1 - \left(1 - \frac{1}{x}\right)^K - \frac{K}{x}\left(1 - \frac{1}{x}\right)^{K-1} =
1 -\left(1 + \frac{K-1}{x}\right)\left(1 - \frac{1}{x}\right)^{K-1} . $$
The derivative approaches $1$. In particular, when $x$ is large enough, this derivative is at least $1/2$ (say). Therefore the function grows to infinity with $x$.
Edit: In fact, as André points out, the derivative approaches $0$, and what is even worse, it's not even clear whether it remains positive. In fact, approximately $(1-1/x)^{K-1} = 1-(K-1)/x+(K-1)(K-2)/2x^2$ and so the derivative is approximately
$$1 - \left(1 + \frac{K-1}{x}\right)\left(1 - \frac{K-1}{x} + \frac{(K-1)(K-2)}{2x^2}\right) = \frac{K(K-1)}{2x^2} + \frac{(K-1)^2(K-2)}{2x^3}.$$
So it does appear that the function is ever-increasing. At the limit, we get the value $K$, as mentioned in the comments.
