By Herbert S. Wilf
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Extra info for Algorithms and Complexity (Second edition)
Just to exercise √ skills in asymptotics, let’s observe √ our newly acquired that since (1 + 5)/2 > 1 and |(1 − 5)/2| < 1, it follows that when n is large we have √ ¶n+1 µ 1 1+ 5 Fn ∼ √ . 2 5 The process of looking for a solution in a certain form, namely in the form αn , is subject to the same kind of special treatment, in the case of repeated roots, that we find in diﬀerential equations. Corresponding to a double root α of the associated quadratic equation α2 = aα + b, we would find two independent solutions αn and nαn , so the general solution would be in the form αn (c1 + c2 n).
A cycle is a circuit if v1 is the only repeated vertex in it. We may say that a circuit is a simple cycle. We speak of Hamiltonian and Eulerian circuits of G as circuits of G that visit, respectively, every vertex, or every edge, of a graph G. There is a world of diﬀerence between Eulerian and Hamiltonian paths, however. If a graph G is given, then thanks to the following elegant theorem of Euler, it is quite easy to decide whether or not G has an Eulerian path. In fact, the theorem applies also to multigraphs, which are graphs except that they are allowed to have several diﬀerent edges joining the same pair of vertices.
X xk = ex = k=0 sin x = cos x = log 1 1−x = 1 1−x ∞ X xm m! 13) (−1)r x2r+1 (2r + 1)! 14) (−1)s x2s (2s)! 16) Can you find a simple form for the sum (the logarithms are ‘natural’) 1 + log 2 + (log 2)3 (log 2)2 + + ··· ? 2! 3! ) Aside from merely substituting values of x into known series, there are many other ways of using known series to express sums in simple form. Let’s think about the sum: 1 + 2 · 2 + 3 · 4 + 4 · 8 + 5 · 16 + · · · + N 2N −1 . 17) is a little diﬀerent because of the multipliers 1, 2, 3, 4, .
Algorithms and Complexity (Second edition) by Herbert S. Wilf