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:

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

Proof

\(\ds z_n\)

\(=\)

\(\ds \tan i n\)

\(\ds \)

\(=\)

\(\ds i \dfrac {1 - e^{2 i \paren {i n} } } {1 + e^{2 i \paren {i n} } }\)

Euler's Tangent Identity

\(\ds \)

\(=\)

\(\ds i \dfrac {1 - e^{-2 n} } {1 + e^{-2 n} }\)

$i^2 = 1$

\(\ds \)

\(\to\)

\(\ds i \dfrac {1 - 0} {1 + 0}\)

$e^{-2 n} \to 0$ as $n \to \infty$

\(\ds \)

\(=\)

\(\ds i\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $1 \ \text {(v)}$