Category:Examples of Convergent Complex Sequences

This category contains examples of Convergent Complex Sequence.

Definition 1

Let $\sequence {z_k}$ be a sequence in $\C$.

$\sequence {z_k}$ converges to the limit $c \in \C$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

where $\cmod z$ denotes the modulus of $z$.

Definition 2

Let $\sequence {z_k} = \sequence {x_k + i y_k}$ be a sequence in $\C$.

$\sequence {z_k}$ converges to the limit $c = a + i b$ if and only if both:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$

where $\size {x_n - a}$ denotes the absolute value of $x_n - a$.