Convergent Complex Sequence/Examples

Examples of Convergent Complex Sequence

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:

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