Solns_ProblemsGeneratingFcns

# Solns_ProblemsGeneratingFcns - Stat 333 Winter 2007...

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Unformatted text preview: Stat 333 Winter 2007 Solutions: Generating Functions #1 (a) ∞ n =0 n 5 s n . Use a ratio test (other tests are also possible). lim n → ∞ | ( n + 1) 5 s n +1 n 5 s 5 | = lim n → ∞ | ( n + 1 n ) 5 s | = | s | So this limit is < 1 iff | s | < 1 i.e. converges for | s | < 1 (b) ∞ n =0 1 (log( n + 2)) n s n = ∞ n =0 [ s log( n + 2) ] n . Now let s be any real number. Then there exists an integer N such that n > N ⇒ | s log( n +2) | < 1 (because s is fixed and log( n + 2) is growing larger). Thus the tail ∞ n = N +1 [ s log( n + 2) ] n is a convergent geometric series. Thus the original series converges for all values of s . (c) Fix any s > 0. Then ns > 1 for all integers n > 1 /s . It follows that ∑ n ( ns ) n ≥ ∑ n> 1 /s 1 = ∞ . Therefore the series only converges at s = 0 so the generating function does not exist. #2 Note that 1 1- x = ∑ ∞ n =0 x n and so 1 (1- x ) 2 = ∞ n =0 x n · ∞ m =0 x m = ∞ k =0 c k x k , where c k = ∑ k n =0 1 · 1 = k +1. In the above calculation, we use the formula for the convolution+1....
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Solns_ProblemsGeneratingFcns - Stat 333 Winter 2007...

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