0809stirling

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Unformatted text preview: O.H. Probability and Markov Chains MATH 2561/2571 E09 1 Asymptotics of n! For the study of random walks on the lattice it was important to know the asymptotics of n!. The first result is given by the following lemma. Lemma 1.1 Denote bn = log(n!) = x log x - x n 1 n k=1 log k. Then n 1 bn x log x - x) + log n . Proof. Let f (x) be an increasing function with f (1) = 0; by comparing the areas, we have (see the picture!) 11111 00000 11111 00000 1111111111 0000000000 1111111111 0000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 11111 00000 11111111111111111111 00000000000000000000 111111111111111 000000000000000 11111 00000 1111111111 0000000000 1111111111 0000000000 111111111111111 000000000000000 111111111111111 000000000000000 11111111111...
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