Convergent Complex Sequence/Examples/((1 + i n) over (1 + n))^3

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 $1 \ \text {(i)}$