Convergence criterion

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In analysis , a convergence criterion is a criterion that can be used to prove the convergence of a sequence or series . In particular, this means criteria for the convergence of real sequences or series. Some of these criteria can also be used to demonstrate the divergence of a sequence or series.

Convergence criteria for consequences

Important convergence criteria for consequences are:

  • Monotony criterion : A monotonic sequence of real numbers converges if and only if it is bounded.
  • Cauchy criterion : A sequence of real or complex numbers converges if and only if it is a Cauchy sequence .
  • Sandwich criterion : A sequence of real numbers converges if it can be estimated downwards and upwards by convergent sequences that have the same limit value.

Convergence criteria for series

Decision tree for determining the convergence behavior of series.

There are three types of convergence criteria for series:

  • Direct criteria that infer convergence from properties of the partial sum sequence of the series,
  • Comparison criteria of the first type, which compare the absolute value or the norm of the series members with a known series, and
  • Comparison criteria of the 2nd type, which compare the quotients of the absolute amounts of successive terms with the corresponding quotients of a known series.

The following table provides an overview of known convergence criteria. The criteria allow different statements: some only allow the conclusion of convergence, with others also divergence can be proven, some show absolute convergence , others only convergence (from absolute convergence follows convergence, but not vice versa). In addition, various criteria allow an estimate of the limit value or an error estimate.

criteria only for
monotonous episodes
convergence divergence absolute
appraisal Error
Zero sequence criterion x Direct criterion
Monotony criterion x x
Leibniz criterion x x x x
Cauchy criterion x x
Abel criterion x x
Dirichlet criterion x x
Majorant criterion x x Comparison criterion
1. Art
Minorant criterion x
Root criterion x x x x
Integral criterion x x x x x
Cauchy compression criterion x x x x
Limit value criterion x x
Quotient criterion x x x x Comparison criterion
2. Art
Gaussian criterion x x x
Raabe criterion x x x
Grief criterion x x x
Bertrand criterion x x x
Ermakoff criterion x x x x

See also