Will this cookie last forever?
A child is given a delicious chocolate chip cookie and eagerly takes a huge bite, devouring exactly of it. As he munches, he decides to try to make his cookie last a while. So, instead of eating the last half, he eats only half of the remaining half, or of his cookie. Then, he eats half of this fourth, or of the cookie. If he continues in this way, always eating exactly half of his remaining cookie, will he ever finish it completely?
Assume that he is always able to precisely divide his cookie in half regardless of how small it is, and that each division takes the same amount of time to eat.
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1 solution
We can describe the amount of cookie left as a series:
2 1 , 4 1 , 8 1 , 1 6 1 , . . .
While the limit of this series is equal to 0 , or
lim x → ∞ = 0
we know that the boy can always continue to divide the cookie in half, regardless of how small it gets. Thus, he will never reach this limit, and the cookie will never be finished completely!