536 An Introduction to Stochastic Modeling From Taylor's formula with remainder (s) = 1 - s 2 + 2 1 (s), 1 (s) s0 s 2 lim = 0 (11.20) e - 2 = 1 - s 2 s 2 + 2 2 (s), s0 lim 2 (s) s 2 = 0. (11.21) The characteristic function of the normalized sum is n (t) := E e S it n n = E e it 1 n Y n t = n n The characteristic function of the standard normal distribution is e -t /2 , so that the difference is written using the identity A n - B n = (A - B) A n-1 + · · · + B n-1 : 2 n (t) - e -t 2 /2 t = n t = n n - e -t - e -t 2 /2n n 2 /2n A n-1 + · · · + B n-1 2 where A = (t/ n), B = e -t /2n . Each of the n terms on the right is less than 1 in modulus, so that we can write n (t) - e -t 2 /2 t 2 n - e -t /2n . n Setting s = t/ n in (11.20) and (11.21) with t fixed, we have t n = 1 - t 2 + 2n 1 t n , e -t 2 /2n = 1 - t 2 + 2n 2 t n . Subtracting these two expressions, the first two terms cancel and we are left with terms of the form n (t/ n), which tend to zero when n and t is fixed. We have proved that n (t) converges to the standard normal characteristic function, which has a continuous distribution function. Hence, by the continuity theorem, the probabilities of all intervals converge, as required. 11.6 Stirling's Formula and Applications Often, we encounter the factorial function of a large integer argument. The numerical evaluation of these expressions can be cumbersome, which leads one to search for an