Definition:Convergent Sequence/Complex Numbers

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Definition

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 $x_n - a$.


Examples

Example: $\dfrac {\paren {3 + i n}^2} {n^2}$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \dfrac {\paren {3 + i n}^2} {n^2}$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = -1$


Example: $\paren {\dfrac {1 + i n} {1 + n} }^3$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \paren {\dfrac {1 + i n} {1 + n} }^3$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = -i$


Example: $\paren {\cos \dfrac \pi n + i \sin \dfrac \pi n}^{2 n + 1}$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \paren {\cos \dfrac \pi n + i \sin \dfrac \pi n}^{2 n + 1}$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = 1$


Example: $\paren {\dfrac 1 2 + i \dfrac 4 5}^n$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \paren {\dfrac 1 2 + i \dfrac 4 5}^n$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = 0$


Example: $\paren {\cos \dfrac \pi {n + 1} + i \sin \dfrac \pi {n + 1} }^n$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \paren {\cos \dfrac \pi {n + 1} + i \sin \dfrac \pi {n + 1} }^n$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = -1$


Example: $\tan i n$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \tan i n$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = i$


Also see