Section 10.1 : Sequences
5. Determine if the given sequence converges or diverges. If it converges what is its limit?
\[\left\{ {\frac{{{{\bf{e}}^{5n}}}}{{3 - {{\bf{e}}^{2n}}}}} \right\}_{n = 1}^\infty \]Show All Steps Hide All Steps
Start SolutionTo answer this all we need is the following limit of the sequence terms.
\[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\bf{e}}^{5n}}}}{{3 - {{\bf{e}}^{2n}}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{5{{\bf{e}}^{5n}}}}{{ - 2{{\bf{e}}^{2n}}}} = \mathop {\lim }\limits_{n \to \infty } \frac{5}{{ - 2}}{{\bf{e}}^{3n}} = -\infty \]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.
Show Step 2We can see that the limit of the terms existed and but was infinite and so we know that the sequence diverges.