All convergent series converge to 1

Convergence and divergence of a series prove: Convergence criteria - Serlo "Math for non-freaks"

Decision tree for convergence and divergence of series

We've cleared that up a number the consequence corresponds to the partial sums. A series converges when the sequence of partial sums converges. Otherwise the series diverges. In the case of convergence, corresponds also the limit of the partial sum sequence.

In this chapter we are concerned with the question of how one can tell whether a series is converging or diverging. To answer this question, there are various convergence criteria that we now consider. We will discuss these convergence criteria in more detail in the next few chapters.

Criteria for convergence

The proofs of the following theorems are dealt with in the main articles of the respective criterion. Given a series . The following criteria are used to determine the convergence of this series:

Absolute convergence [edit]

Main article: Absolute convergence of a range

definition (Absolute convergence)

A row is called absolutely convergent if converges.

sentence (absolute convergence)

The normal convergence of a series follows from the absolute convergence of a series. If so converges, then also converges .

example (absolute convergence)

The series converges because it absolutely converges. The series namely, their absolute values ​​converges.

Cauchy criterion [edit]

Main article: Cauchy criterion for series

sentence (Cauchy criterion)

There is for everyone a , so that for all then the series converges.

example (Cauchy criterion)

The geometric series converges by Cauchy's criterion because it is

Be . There is there is a With for all . For this is after the above transformation too for all . The series thus converges according to the Cauchy criterion.

Leibniz criterion [edit]

Main article: Leibniz criterion

sentence (Leibniz criterion)

When the turn is the shape has and if is a nonnegative monotonically decreasing zero sequence, then the series converges.

example (Leibniz criterion)

The series converges according to the Leibniz criterion because the consequence is a monotonically decreasing zero sequence.

Majorant criterion [edit]

Main articles: Majorant Criterion and Minorant Criterion

sentence (Majorant criterion)

Be for all . If converges, then the series converges absolutely.

example (Majorant criterion)

It is . So that is . Because converges (namely to 1), the series converges . Because all summands are positive, the convergence is absolute.

Quotient criterion [edit]

Main article: Quotient criterion