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:

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


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