Convergent Complex Sequence/Examples
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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:
- $\ds \lim_{n \mathop \to \infty} z_n = -1$
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:
- $\ds \lim_{n \mathop \to \infty} z_n = -i$
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:
- $\ds \lim_{n \mathop \to \infty} z_n = 1$
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:
- $\ds \lim_{n \mathop \to \infty} z_n = 0$
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:
- $\ds \lim_{n \mathop \to \infty} z_n = -1$
Example: $\tan i n$
Let $\sequence {z_n}$ be the complex sequence defined as:
- $z_n = \tan i n$
Then:
- $\ds \lim_{n \mathop \to \infty} z_n = i$