I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 10.4 : Convergence/Divergence of Series
4. 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.
\[{s_n} = \frac{{{n^2}}}{{5 + 2n}}\] 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,
\[\mathop {\lim }\limits_{n \to \infty } {s_n} = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^2}}}{{5 + 2n}} = \infty \]Now, we can see that this limit exists and is infinite. Therefore, we now know that the series, \(\sum\limits_{n = 0}^\infty {{a_n}} \), diverges.
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.