Convergent Complex Sequence/Examples/tan i n

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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 $n \to \infty$
\(\ds \) \(=\) \(\ds i\)

$\blacksquare$


Sources