Zero sequence criterion
The zero sequence criterion , also known as trivial criterion or divergence criterion , is a convergence criterion in mathematics , according to which a series diverges if the sequence of its summands is not a zero sequence . The zero sequence criterion is a necessary but not a sufficient condition for the convergence of a series.
criteria
The zero sequence criterion is:
- If the sequence of the summands of a series does not form a zero sequence, then the series diverges.
So it applies to the summands of a series
or if this limit does not exist, the series does not converge. In contrast to other convergence criteria , the zero sequence criterion can only be used to prove that a series is diverging, but not to decide whether it converges. For example, the harmonic series does not converge even though its summands form a zero sequence.
Examples
The series
diverges, because
- .
also diverges because the limit value
does not exist.
proof
The proof of the zero sequence criterion is typically done by counterposition , that is, by reversing the statement
- If a series converges, the sequence of the summands forms a zero sequence.
A series converges if the sequence of its partial sums with
converges, that is, there is a limit such that
- .
By rearranging the series and using the calculation rules for limit values then applies
Since the sequence of summands has to form a zero sequence for every convergent series, a series diverges if this is not the case.
Alternative proof using the Cauchy criterion
The trivial criterion can also be proven using the Cauchy criterion . According to this criterion, a series converges if and only if there is a minimum index for all such that is for all . If we put here , it follows: For all there is one , so that the inequality is fulfilled for all . But this is exactly the definition that the sequence is a null sequence.
See also
- Olivier's theorem , a tightening of the zero sequence criterion
literature
- Oliver Deiser: Analysis 1, Volume 1 . Springer, 2011, ISBN 3-642-22459-8 .
- Wolfgang Walter : Analysis, Volume 1 . Springer, 2006, ISBN 3-540-35078-0 .
Web links
- Todd Rowland: Limit Test . In: MathWorld (English).