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limit of nth root of n

sequence of nth roots of n
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Mathematics Subject Classification

12D99 no label found30-00 no label found


Here is a puzzle:

Let's say someone shows you a rod 99 feet in length.
He tells you that you are allowed to cut it up as you see, and you will be given the product of the lengths of all of the pieces in dollars.
For example, if you were to slice it in two pieces 48.5 feet in length, you would be given 48.5 x 48.5 dollars, (2352.25)

How should you cut up the rod?

The answer: make 33 pieces each 3 feet long.
The reason: the nth root of n reaches it's maximum at 3.

It is true that the maximum is reached for all pieces having the same length. Then, the product for n pieces is (S/n)^n (S=99). The maximum is reached at n = S/e = 36.42; the closest integer is 36, not 33.

Cut one piece an 1/8'' long and another piece 98' 11 7/8'' long.
Then I would pay a dollar. If that is still too much for you,
make the short piece shorter.

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