How much larger is the AM than the GM?

Calculus Level 3

From the AM-GM inequality, we know that the arithmetic mean (AM) of a list of non-negative real numbers is always greater than or equal to the geometric mean (GM). But the inequality doesn't tell us just how much larger the AM is.

Given a list of nn random real numbers chosen uniformly and independently in the range [a0,a1],[a_0,a_1], where 0a0<a1,0\le a_0<a_1, find E[AMGM]\text{E}\left[\frac{\text{AM}}{\text{GM}}\right] in terms of n,a0,a1.n,a_0,a_1.

Then find lima1a00E[AMGM]\displaystyle \lim\limits_{\overset{a_0 \to 0}{a_1 \to \infty}} \text{E}\left[\frac{\text{AM}}{\text{GM}}\right].

If the formula is of the form nn(AnB)(n1)nC,\large\frac{n^n}{\left(An-B\right){\left(n-1\right)}^{n-C}}, where A,B,CA,B,C are positive integers, give your answer as A+B+C.A+B+C.

Note: The notation E[X]\text{E}[X] is the expected value of the random variable X.X.


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