Section 10.4 : Convergence/Divergence of Series
3. Assume that the \(n\)th term in the sequence of partial sums for the series \(\displaystyle \sum\limits_{n = 0}^\infty {{a_n}} \) is given below. Determine if the series \(\displaystyle \sum\limits_{n = 0}^\infty {{a_n}} \) is convergent or divergent. If the series is convergent determine the value of the series.
\[{s_n} = \frac{{5 + 8{n^2}}}{{2 - 7{n^2}}}\] Show SolutionThere really isn’t all that much that we need to do here other than to recall,
\[\sum\limits_{n = 0}^\infty {{a_n}} = \mathop {\lim }\limits_{n \to \infty } {s_n}\]So, to determine if the series converges or diverges, all we need to do is compute the limit of the sequence of the partial sums. The limit of the sequence of partial sums is,
\[\mathop {\lim }\limits_{n \to \infty } {s_n} = \mathop {\lim }\limits_{n \to \infty } \frac{{5 + 8{n^2}}}{{2 - 7{n^2}}} = - \frac{8}{7}\]Now, we can see that this limit exists and is finite (i.e. is not plus/minus infinity). Therefore, we now know that the series, \(\sum\limits_{n = 0}^\infty {{a_n}} \), converges and its value is,
\[\sum\limits_{n = 0}^\infty {{a_n}} = - \frac{8}{7}\]If you are unfamiliar with limits at infinity then you really need to go back to the Calculus I material and do some review of limits at infinity and L’Hospital’s Rule as we will be doing quite a bit of these kinds of limits off and on over the next few sections.