Convergent Real Sequence/Examples/n^3 + 5 n^2 + 2 over 2 n^3 + 9
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Example of Convergent Real Sequence
- $\ds \lim_{n \mathop \to \infty} \paren {\dfrac {n^3 + 5 n^2 + 2} {2 n^3 + 9} } = \dfrac 1 2$
Proof
\(\ds \dfrac {n^3 + 5 n^2 + 2} {2 n^3 + 9}\) | \(=\) | \(\ds \dfrac {1 - \dfrac 5 n + \dfrac 2 {n^3} } {2 + \dfrac 9 {n^3} }\) | dividing top and bottom by $n^3$ | |||||||||||
\(\ds \) | \(\to\) | \(\ds \dfrac {1 + 0 + 0} {2 + 0}\) | \(\ds \text {as $n \to \infty$}\) | Sequence of Powers of Reciprocals is Null Sequence | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2\) |
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.20 \ (1)$