Convergent Real Sequence/Examples/2 n^3 - 3 n over 5 n^3 + 4 n^2 - 2
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Example of Convergent Real Sequence
- $\ds \lim_{n \mathop \to \infty} \paren {\dfrac {2 n^3 - 3 n} {5 n^3 + 4 n^2 - 2} } = \dfrac 2 5$
Proof
\(\ds \dfrac {2 n^3 - 3 n} {5 n^3 + 4 n^2 - 2}\) | \(=\) | \(\ds \dfrac {2 - \dfrac 3 {n^2} } {5 + \dfrac 4 n - \dfrac 2 {n^3} }\) | dividing top and bottom by $n^3$ | |||||||||||
\(\ds \) | \(\to\) | \(\ds \dfrac {2 - 0} {5 + 0 - 0}\) | \(\ds \text {as $n \to \infty$}\) | Sequence of Powers of Reciprocals is Null Sequence | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 2 5\) |
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.9$: Example