Convergence of Series Using Comparison Test

Determine whether the series $\displaystyle \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}$ is convergent using the comparison test.

In this problem, we are tasked with determining the convergence of an infinite series using the comparison test, which is a fundamental tool in analyzing the behavior of infinite series in calculus. The comparison test involves comparing a given series with a second series whose convergence properties are known. Specifically, if you can find a convergent series that is greater than or equal to the series in question, then the original series also converges. Conversely, if you can find a divergent series that is smaller than or equal to the series in question, then the original series diverges.

In this specific problem, the series presented has the general term involving a rational function of n, which suggests that we might compare it to a p-series or another series with a similar form that has established convergence properties. Understanding the growth of the terms as n becomes very large is crucial in selecting an appropriate comparative series. The problem tests your ability to recognize patterns in the terms and to use inequalities effectively to bound the terms, which is essential for applying the comparison test effectively.

The comparison test, while straightforward in its logic, requires careful consideration of the inequalities involved and sometimes a bit of creativity in choosing the series for comparison. It's a powerful technique not only in determining convergence but also in enhancing understanding of how series behave, more generally, which is why it's a staple concept in any calculus course focusing on sequences and series.

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