Suppose we want to estimate a sum
where the sequence is a monotone sequence of non negative real numbers and are real numbers. Later we can let for example and likewise but let us keep finite for the moment. In this case is a standard tactic to look for a real function with the same kind of monotonicity as the sequence , such that for all the we are interested in. Then we have
Usually the choice of the function should be obvious by looking at the sequence . If for example the sequence is explicitly given then the first obvious choice would be to replace the discrete parameter with a continuous variable and look at the resulting function .
Let us look at the partial sums of the harmonic series
The sequence is a strictly decreasing sequence of positive numbers. Taking we recover the well known estimate
One can use the same technique to prove that for a positive real number , the over-harmonic series
converge exactly when and we have the estimate