This is a simplified inventory problem. Suppose that it costs $c$ dollars to stock an item and that the item sells for $s$ dollars. Suppose that the number of items that will be asked for by customers is a random variable with the frequency function $p(k)$. Find a rule for the number of items that should be stocked in order to maximize the expected income. (Hint: Consider the difference of successive terms.)
The answer given by the textbook is: That value of $n$ such that $s \sum_{k=n}^{\infty} p(k) > c \sum_{k=1}^{n-1} p(k) $ and $\sum_{k=n+1}^{\infty} p(k) > c \sum_{k=1}^{n} p(k)$.
I don't really understand why that is. Any help is appreciated.