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:
- $\ds \lim_{n \mathop \to \infty} z_n = -1$
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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Example $\text {(i)}$