proposition for any real number there is also combination of the two common formscalled the linear exponential summationwhich has the following expansionproposition for the nth harmonic number is defined as proposition if is the nth harmonic numberthen is ln th( basic probability we review some basic facts from probability theory the most basic is that any statement about probability is defined upon sample space swhich is defined as the set of all possible outcomes from some experiment we leave the terms "outcomesand "experimentundefined in any formal sense example consider an experiment that consists of the outcome from flipping coin five times this sample space has different outcomesone for each different ordering of possible flips that can occur sample spaces can also be infiniteas the following example illustrates example consider an experiment that consists of flipping coin until it comes up heads this sample space is infinitewith each outcome being sequence of tails followed by single flip that comes up headsfor , , probability space is sample space together with probability function pr that maps subsets of to real numbers in the interval [ , it captures mathematically the notion of the probability of certain "eventsoccurring formallyeach subset of is