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 $\displaystyle \lim_{n \mathop \to \infty} z_n$ 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