Convergent Complex Sequence/Examples/(3+in)^2 over n^2

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Example of Convergent Complex Sequence

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \dfrac {\paren {3 + i n}^2} {n^2}$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = -1$


Proof

\(\ds z_n\) \(=\) \(\ds \dfrac {\paren {3 + i n}^2} {n^2}\)
\(\ds \) \(=\) \(\ds \dfrac {9 + 6 i n - n^2} {n^2}\)
\(\ds \) \(=\) \(\ds \dfrac {9 - n^2} {n^2} + \dfrac {6 i} n\)
\(\ds \) \(=\) \(\ds \dfrac 9 {n^2} - 1 + i \dfrac 6 n\)
\(\ds \) \(\to\) \(\ds 0 - 1 + i \times 0\) as both $\sequence {\dfrac 9 {n^2} }$ and $\sequence {\dfrac 6 n}$ are Basic Null Sequences

$\blacksquare$


Sources