Log Log Log
Thu, Apr 14, 2016 02:00 CEST
Theorem
loglognlogn!=Θ(n)
Proof
K2!K2→∞limK2!n→∞limK2!=loglognlogn2!=loglognlogn!=2πloglognlogn(eloglognlogn)loglognlogn=2πloglognlogn(eloglognlogn)loglognlogn=2πloglognlogn(eloglognlogn)loglognlogn=2πloglognlogn2logloglognloglognlogn2logeloglognlogn2loglognloglognlogn=2πloglognlognnloglognlogloglognnloglognlogen=2π2logloglogn2loglognnloglognlogloglognnloglognlogen=2πnlognlogloglognnlognloglognnloglognlogloglognnloglognlogen=2πn2lognlogloglognn2lognloglognnloglognlogloglognnloglognlogen=2πn2lognlogloglognn2lognloglognnloglognlogloglogn+logen=2πn2lognloglognloglognlogloglognn2lognloglogn(loglogn)2n2lognloglogn2lognlogloglogn+2lognlogen=2πn2lognloglogn2logelogn+2lognlogloglogn+loglognlogloglogn−(loglogn)2n=2πn2lognloglogn2lognlogloglognn=2πnloglognlogloglognn=2πn1−loglognlogloglogn