Calculus II - Sequences

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Section 10.1 : Sequences

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3. Determine if the given sequence converges or diverges. If it converges what is its limit?

\[\left\{ {\frac{{{n^2} - 7n + 3}}{{1 + 10n - 4{n^2}}}} \right\}_{n = 3}^\infty \]

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Start Solution

To answer this all we need is the following limit of the sequence terms.

\[\mathop {\lim }\limits_{n \to \infty } \frac{{{n^2} - 7n + 3}}{{1 + 10n - 4{n^2}}} = - \frac{1}{4}\]

You do recall how to take limits at infinity right? If not you should go back into the Calculus I material do some refreshing on limits at infinity as well at L’Hospital’s rule.

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We can see that the limit of the terms existed and was a finite number and so we know that the sequence converges and its limit is \( - \frac{1}{4}\).