Divergent Complex Sequence/Examples/(-1)^n + 1 over n
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Example of Divergent Complex Sequence
Let $\sequence {z_n}$ be the complex sequence defined as:
- $z_n \paren {-1}^n + \dfrac i n$
Then $\ds \lim_{n \mathop \to \infty} z_n$ does not exist.
Proof
The real part of $z_n$ is $\paren {-1}^n$.
As can be seen in Divergent Sequence may be Bounded, $\sequence {\paren {-1}^n}$ does not converge to a limit.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Example $\text {(iii)}$