Expected value for a profit

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.

I would think the expected income would be

[; (s-c) \sum_{k=1}^{\infty} k p(k) ;].

The answer is apparently the number n such that

[; s \sum_{k=n}^{\infty} p(k) > c \sum_{k=1}^{n-1} p(k) ;]

and

[; s \sum_{k=n+1}^{\infty} p(k) < c \sum_{k=1}^n p(k) ;].

This sort of makes sense, but I don't really see how to manipulate the E(X) sum into these sums. Any help would be appreciated. Thanks.