proposition if > is an integer constantthen ik is th(nk+ = another common summation is the geometric sumni= ai for any fixed real number  proposition an+ ai = for any real number  proposition ai = for any real number there is also combination of the two common formscalled the linear exponential summationwhich has the following expansionproposition for iai = ( ) ( + na( + ( ) the nth harmonic number hn is defined as hn = proposition if hn is the nth harmonic numberthen hn 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