Convergence of Series with Polynomial Terms

Determine the convergence or divergence of the series {n=1}^{\infty} \frac{n+5}{n^2}.

When addressing the convergence or divergence of a series, it's crucial to assess the behavior of its terms as the series progresses. For the given series, which is the sum of rac{n+5}{n^2} from $n = 1$ to infinity, the essence lies in understanding how to apply established convergence tests. One may initially consider comparing this series to a known benchmark series by focusing on the dominant term in the fraction as $n$ approaches infinity, which in this case is $1/n$. This leads us to believe that the series could behave like the harmonic series, which is known to diverge.

A powerful technique in assessing the convergence is the Comparison Test, where you compare your series to another series with known behavior. In this instance, a direct comparison to the harmonic series itself helps in determining the behavior of the original series. Alternatively, the Limit Comparison Test might also be useful, as it considers the limit of the ratio of the terms from one series to a reference series.

Additionally, understanding polynomial asymptotic behavior as $n$ becomes very large allows one to isolate the leading term's impact on convergence. The Divergence Test, which states that if the limit of the sequence of terms of the series does not tend to zero, the series diverges, may seem applicable but can lead to inconclusive results if the terms actually approach zero without providing further insights into convergence. Therefore, grasping these high-level strategies offers a robust framework for approaching and solving problems related to the convergence or divergence of series.