Theoretical Computer Science Cheat SheetDefinitionsSeriesf (n) = (g(n))iff f (n) = O(g(n)) andf (n) = (g(n)).f (n) = o(g(n))iff limn f (n)/g(n) = 0.iff > 0, n0 such that|an a| < , n n0 .lim an = anleast b R such that b s,s S.sup Sgreatest b R such that b s, s S.inf Slim inf{ai | i n, i N}.lim inf ani=1n(n + 1),2nXi2 =i=1nXn(n + 1)(2n + 1),6i3 =i=1n2 (n + 1)2.4In general:nnXX1(n + 1)m+1 1 (i + 1)m+1 im+1 (m + 1)imim =m+1i=1i=1n1mX1 X m+1Bk nm+1k .im =km+1i=1omiff positive c, n0 such thatf (n) cg(n) 0 n n0 .i=k=0Geometric series:nXcn+1 1,ci =c1i=0nXici =i=0c 6= 1,Xci =i=0ncn+2 (n + 1)cn+1 + c,(c 1)2Harmonic series:nX1,Hn =ii=11,1cXnXc 6= 11,,ci =i=1i=1X.cf (n) = (g(n))nXici =i=0ceiff positive c, n0 such that0 f (n) cg(n) n n0 .f (n) = O(g(n))c,1cc,(1 c)2|c| < 1,|c| < 1.n( + 1)n(nn(n 1)n(Hn .24lim sup anlim sup{ai | i n, i N}.i=1i=1nnnn XXin+11nCombinations:SizeksubHi =Hn+1 .Hi = (n + 1)H1)Hn n,kmm+1m+1sets of a size n set.i=1i=1n n XStirling numbers (1st kind):n!nn!nnnkn,=2 ,==,2.1.3.Arrangements of an n el ...
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