Almost Pi

Let $f_n(k) = e^{k/n} – 1$, for all non-negative integers $k$. Remarkably, $f_{200}(6)+f_{200}(75)+f_{200}(89)+f_{200}(226)=\underline{3.1415926}44529\cdots\approx\pi$. In fact, it is the best approximation of $\pi$ of the form $f_n(a) +...
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