First note that
we can assume without loss of generality that the series will start at as we’ve done for all our series test
proofs. Also note that this proof is
very similiar to the proof of the Ratio Test.
Let’s start off
the proof here by assuming that and we’ll need to show that is absolutely convergent. To do this let’s first note that because there is some number r such that .
and because we
also have chosen r such that there is some N such that if we will have,
Now the series
is a geometric
series and because we in fact know that it is a convergent
series. Also because by the Comparison test the series
convergent. However since,
we know that is also convergent since the first term on
the right is a finite sum of finite terms and hence finite. Therefore is absolutely convergent.
Next, we need
to assume that and we’ll need to show that is divergent. Recalling that,
and because we know that there must be some N such that if we will have,
However, if for all then we know that,
This in turn
the Divergence Test is divergent.
need to assume that and show that we could get a series that has
any of the three possibilities. To do
this we just need a series for each case.
We’ll leave the details of checking to you but all three of the
following series have and each one exhibits one of the