User:Natsuhata/Draft

https://en.wikipedia.org/wiki/Perpetual_stew

Mathematical model of average age
Let $D\in\mathbb{N}$ be the age of the perpetual stew in days, and let $$p\in[0,1)$$ the percentage (where $$p=0.17$$ equals to 17 %) of stew left in the pot after every day. The stew is filled with fresh ingredients and stirred thoroughly at the beginning of each day. Then the average age (in days) of the stew at the time of the refilling is given by the partial sum  $$A_{p,D}=\sum_{d=0}^D d\,p^d=\frac {D\,p^{D+2} - (D+1)\,p^{D+1} + p}{(1-p)^2},$$whose age limit $A_p$  with respect to $D$  is given as the series$$A_p=\lim_{D\to\infty} A_{p,D}=\sum_{d=0}^\infty d\,p^d=\frac {p}{(1-p)^2}.$$The partial sum $A_{p,D}$  consists of nonnegative summands, hence increases as $D$  or $p$  is increasing, and $A_p$  is the limit and an upper bound for $D\to\infty$, hence $A_{p,D}\leq A_{p}$ . The limit $\lim_{p\to1} A_{p}=\infty$  tends to infinity and behaves as naively expected. Naturally, $A_{p,D}$ and $A_{p}$  describe upper bounds if $$p\in[0,1)$$ is an upper bound for the amount of stew left in the pot after every day. Trivially the continuation is $A_{1,D}=D $.