Convergent Real Sequence/Examples/n^3 + 5 n^2 + 2 over 2 n^3 + 9

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 $\S 4.20 \ (1)$