Cheat Sheet

    Abstract

    Definitions Series f (n) = O(g(n)) iff ∃ positive c, n 0 such that 0 ≤ f (n) ≤ cg(n) ∀n ≥ n 0. n i=1 i = n(n + 1) 2 , n i=1 i 2 = n(n + 1)(2n + 1) 6 , n i=1 i 3 = n 2 (n + 1) 2 4. In general: n i=1 i m = 1 m + 1 (n + 1) m+1 − 1 − n i=1 (i + 1) m+1 − i m+1 − (m + 1)i m n−1 i=1 i m = 1 m + 1 m k=0 m + 1 k B k n m+1−k. Geometric series: n i=0 c i = c n+1 − 1 c − 1 , c = 1, ∞ i=0 c i = 1 1 − c , ∞ i=1 c i = c 1 − c , |c| < 1, n i=0 ic i = nc n+2 − (n + 1)c n+1 + c (c − 1) 2 , c = 1, ∞ i=0 ic i = c (1 − c) 2 , |c| < 1. Harmonic series: H n = n i=1 1 i , n i=1 iH i = n(n + 1) 2 H n − n(n − 1) 4. n i=1 H i = (n + 1)H n − n, n i=1 i m H i = n + 1 m + 1 H n+1 − 1 m + 1. f (n) = Ω(g(n)) iff ∃ positive c, n 0 such that f (n) ≥ cg(n) ≥ 0 ∀n ≥ n 0. f (n) = Θ(g(n)) iff f (n) = O(g(n)) and f (n) = Ω(g(n)). f (n) = o(g(n)) iff lim n→∞ f (n)/g(n) = 0. lim n→∞ a n = a iff ∀ > 0, ∃n 0 such that |a n − a| < <, ∀n ≥ n 0. sup S least b ∈ R such that b ≥ s, ∀s ∈ S. inf S greatest b ∈ R such that b ≤ s, ∀s ∈ S.

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    @inproceedings{CheatS, title={Cheat Sheet}, author={} }