Term test – Serlo
In this chapter we will see an easy and useful criterion for divergence. This criterion is called term test or also null-sequence test or divergence test. It says that every series , where is not a null sequence, is divergent. When you invert that, this means that for every convergent series we have that( is a null sequence).
Term test
[Bearbeiten]Theorem (Term test)
If a series is convergent, then is a null sequence. That means, that every series is divergent, if is divergent or .
Example (Term test)
The series is divergent, because is divergent ( and are both accumulation points and thus there is no limit).
Also the series diverges, because .
Warning
The criterion that is a null sequence is only a necessary but no sufficient criterion for the convergences of the series .
This means: from the fact that we cannot follow that converges. For instance the harmonic series is divergent although .
Proof with telescoping sums
[Bearbeiten]Proof (Term test)
Our premise is that is a convergent series. We want to conclude that is a null sequence.
A member of the sequence can be written as the difference between two consecutive partial sums and :
From our premise we know that has a limit . Thus we have
But also , because the limit will not change if we simply shift indexes. Put together we obtain:
We can conclude that must be null sequence.
Proof using Cauchy criterion
[Bearbeiten]Proof (Term test)
We can proof the same result using the Cauchy criterion. As a reminder, every convergent series satisfies the Cauchy criterion:
We don't consider all , but only the case :
The last formula is exactly the -definition for what it means that is a null sequences. In other words we have showed that .
Example
[Bearbeiten]Exercise
Show that is divergent.
Solution
We have
From the above we see that is not a null sequence. So the series is divergent according to the term test.
Outlook: Stronger version of term test
[Bearbeiten]If we demand that is monotonically decreasing, then we can show that even is a null sequence. See the respective exercise.