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Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.

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Find conditions for the following running average to be monotonically decreasing

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $ …