(Polya, Szego, Problems and Theorems in Analysis). Pick a sequence of positive numbers, , and set . Now write . Show that

where and is the canonical -dimensional simplex. In fact, the limit is .

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# Integrating over simplices, and the Gamma function

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(Polya, Szego, Problems and Theorems in Analysis). Pick a sequence of positive numbers, , and set . Now write . Show that

where and is the canonical -dimensional simplex. In fact, the limit is .

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Don’t let the problem intimidate you. One can verify this is true by carefully organizing a (rather big) sum as a Riemann sum of the given function.

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