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

Mathematical model of average age

[edit]
average age of infinite stew by percentage of daily left stew
stew left in bowl () average age ()
  0 %   0 seconds
  0.1 % 87 seconds
  1 % 15 minutes
10 %   3.0 hours
20 %   7.5 hours
30 % 15 hours
38 % 24 hours
40 % 27 hours
50 %     2 days
56.5 %     3 days
60 %     3.8 days
70 %     7.8 days
72.99 %   10 days
80 %   20 days
90 %   90 days
94.9009 % 365 days
95 % 380 days
99 %          27 years
99.9 %     2,735 years
99.99 % 273,753 years

Let be the age of the perpetual stew in days, and let the percentage (where 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 whose age limit with respect to is given as the seriesThe partial sum consists of nonnegative summands, hence increases as or is increasing, and is the limit and an upper bound for , hence . The limit tends to infinity and behaves as naively expected. Naturally, and describe upper bounds if is an upper bound for the amount of stew left in the pot after every day. Trivially the continuation is .