Math for Non-Geeks/Term test

oben 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 Term test If a series is convergent, then is a null sequence. That means, that every series is divergent, if is divergent or . 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 . 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 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 Term test We can prove 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 sequence. In other words we have showed that . Example thumb|Example exercise for the term test (in German) Show that is divergent. 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 If we demand that is monotonically decreasing, then we can show that even is a null sequence. See the respective exercises. unten