Convergent Complex Sequence/Examples/((1 + i n) over (1 + n))^3
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Example of Convergent Complex Sequence
Let $\sequence {z_n}$ be the complex sequence defined as:
- $z_n = \paren {\dfrac {1 + i n} {1 + n} }^3$
Then:
- $\ds \lim_{n \mathop \to \infty} z_n = -i$
Proof
\(\ds z_n\) | \(=\) | \(\ds \paren {\dfrac {1 + i n} {1 + n} }^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {\frac 1 n + i} {\frac 1 n + 1} }^3\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds \paren {\dfrac i 1}^3\) | as $\dfrac 1 n$ is a Basic Null Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds i^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $1 \ \text {(i)}$