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Section 10.1 : Sequences

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

\[\left\{ {\frac{{\ln \left( {n + 2} \right)}}{{\ln \left( {1 + 4n} \right)}}} \right\}_{n = 1}^\infty \]

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

\[\mathop {\lim }\limits_{n \to \infty } \frac{{\ln \left( {n + 2} \right)}}{{\ln \left( {1 + 4n} \right)}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{}^{1}/{}_{{n + 2}}}}{{{}^{4}/{}_{{1 + 4n}}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{1 + 4n}}{{4\left( {n + 2} \right)}} = 1\]

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

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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 one.