# Convergent Complex Sequence/Examples/tan i n

## Example of Convergent Complex Sequence

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

$z_n = \tan i n$

Then:

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

## Proof

 $\ds z_n$ $=$ $\ds \tan i n$ $\ds$ $=$ $\ds i \dfrac {1 - e^{2 i \paren {i n} } } {1 + e^{2 i \paren {i n} } }$ Tangent Exponential Formulation $\ds$ $=$ $\ds i \dfrac {1 - e^{-2 n} } {1 + e^{-2 n} }$ $i^2 = 1$ $\ds$ $\to$ $\ds i \dfrac {1 - 0} {1 + 0}$ $e^{-2 n} \to 0$ as $n \to \infty$ $\ds$ $=$ $\ds i$

$\blacksquare$