chap mathematical preliminaries ai = an+ for - ( as special cases to equation ni ( = and + ( = as corollary to equation log xn log + ( = finallyn = + ( the sum of reciprocals from to ncalled the harmonic series and written hn has value between loge and loge to be more preciseas growsthe summation grows closer to ( where is euler' constant and has the value most of these equalities can be proved easily by mathematical induction (see section unfortunatelyinduction does not help us derive closed-form solution it only confirms when proposed closed-form solution is correct techniques for deriving closed-form solutions are discussed in section the running time for recursive algorithm is most easily expressed by recursive expression because the total time for the recursive algorithm includes the time to run the recursive call(sa recurrence relation defines function by means of an expression that includes one or more (smallerinstances of itself classic example is the recursive definition for the factorial functionhn loge ( ) for another standard example of recurrence is the fibonacci sequencefib(nfib( fib( for fib( fib(