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Section 10.4 : Convergence/Divergence of Series

6. Show that the following series is divergent.

\[\sum\limits_{n = 5}^\infty {\frac{{6 + 8n + 9{n^2}}}{{3 + 2n + {n^2}}}} \] Show Solution

First let’s note that we’re being asked to show that the series is divergent. We are not being asked to determine if the series is divergent. At this point we really only know of two ways to actually show this.

The first option is to show that the limit of the sequence of partial sums either doesn’t exist or is infinite. The problem with this approach is that for many series determining the general formula for the \(n\)th term of the sequence of partial sums is very difficult if not outright impossible to do. That is true for this series and so that is not really a viable option for this problem.

Luckily enough for us there is actually an easier option to simply show that a series is divergent. All we need to do is use the Divergence Test.

The limit of the series terms is,

\[\mathop {\lim }\limits_{n \to \infty } {a_n} = \mathop {\lim }\limits_{n \to \infty } \frac{{6 + 8n + 9{n^2}}}{{3 + 2n + {n^2}}} = 9 \ne 0\]

The limit of the series terms is not zero and so by the Divergence Test we know that the series in this problem is divergent.