but applying the inequality presented above we have the following the term comes from the last summation from to from the inequality above there are ones that are part of the summation these can be factored out as = log < log ( <(log log ( = = = we now have lower and upper bound for our sum the same summation appears in both the lower and upper bound but what does = log equalthe following equivalences will help in determining this summation ln ln to determine what the summation above is equal to we can establish couple of inequalities that bound the sum from above and below in fig the summation can be visualized as the green area the first term in the summation would provide the first green rectanglethe second green rectangle corresponds to the second term in the summation and so on the black line in the figure is the plot of the log base of clearly the area covered by the green rectangles is bigger than the area under log ln ln log fig plot of log(nthis copy belongs to 'acha