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Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. [1]
The convergence of a geometric series can be described depending on the value of a common ratio, see § Convergence of the series and its proof. Grandi's series is an example of a divergent series that can be expressed as 1 − 1 + 1 − 1 + ⋯ {\displaystyle 1-1+1-1+\cdots } , where the initial term is 1 {\displaystyle 1} and the common ratio ...
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.
In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
Proof: (sequential compactness implies closed and bounded) Suppose A {\displaystyle A} is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the property that every sequence in A {\displaystyle A} has a subsequence converging to an element of A {\displaystyle A} .
The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the Cauchy criterion for convergence of sequences : a sequence of real numbers is convergent if and only if it is a Cauchy sequence.
The proof can also be based on Fatou's lemma instead of a direct proof as above, because Fatou's lemma can be proved independent of the monotone convergence theorem. However the monotone convergence theorem is in some ways more primitive than Fatou's lemma.