Term test

The zero sequence criterion, even trivial criterion or divergence criterion is in mathematics, a convergence criterion by which diverges a number, if the sequence of its summands is not a null sequence. The zero sequence criterion thus constitutes a necessary but not a sufficient condition for the convergence of a series.

Criterion

The zero sequence criterion is:

Consequently, where the summands of a series

Or this limit does not exist, not the series converges. Unlike other convergence criteria can only be proved with the zero sequence criterion that a series diverges and not whether it converges. For example, the harmonic series converges not, although their summands form a null sequence.

Examples

The series

Diverges because

The alternating series

Also diverges, because the limit

Does not exist.

Evidence

The proof of the zero-sequence criterion is typically done by contraposition, ie by reversing the statement

A series converges if the sequence of its partial sums with

Converges, that is to say, there is a limit value, so that

Applies by changing the number and the calculation rules for limits then

After the result of the addend for each convergent series must form a zero sequence diverges a number, if this is not the case.