Definition:Convergence
Definition
Convergence is the property of being convergent, which is defined variously according to the scope of the object in question.
As follows:
Convergent Sequence
Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.
Then $\sequence {x_n}$ converges to the limit $\alpha \in S$ if and only if:
- $\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$
Convergent Series
Let $\struct {S, \circ, \tau}$ be a topological semigroup.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
This series is said to be convergent if and only if its sequence of partial sums $\sequence {s_N}$ converges in the topological space $\struct {S, \tau}$.
If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
Convergent Mapping
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $c$ be a limit point of $M_1$.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ except possibly at $c$.
Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$.
Then $f$ converges to the limit $L$ as $x$ tends to $c$.
Convergent Filter
Let $\struct {S, \tau}$ be a topological space.
Let $\FF$ be a filter on $S$.
Then $\FF$ converges to a point $x \in S$ if and only if:
- $\forall N_x \subseteq S: N_x \in \FF$
where $N_x$ is a neighborhood of $x$.
That is, a filter converges to a point $x$ if and only if every neighborhood of $x$ is an element of that filter.
If there is a point $x \in S$ such that $\FF$ converges to $x$, then $\FF$ is convergent.
Convergent Filter Basis
Let $\struct {S, \tau}$ be a topological space.
Let $\BB$ be a filter basis of a filter $\FF$ on $S$.
Then $\BB$ converges to a point $x \in S$ if and only if:
- $\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$
where $N_x$ is a neighborhood of $x$.
That is, a filter basis is convergent to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.
Convergent Continued Fraction
Let $\struct {F, \norm {\,\cdot\,} }$ be a valued field.
Let $C = \sequence {a_n}_{n \mathop \ge 0}$ be a infinite continued fraction in $F$.
Then $C$ converges to its value $x \in F$ if and only if the following hold:
- $(1): \quad$ For all natural numbers $n \in \N_{\ge 0}$, the $n$th denominator is nonzero
- $(2): \quad$ The sequence of convergents $\sequence {C_n}_{n \mathop \ge 0}$ converges to $x$.
Convergent Integral
A convergent integral is an improper integral whose limits converge to a definite value.
Convergent Iteration
A convergent iteration is an iteration which generates a convergent sequence.
Also see
- Results about convergence can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convergence
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convergence