Limit Comparison Test for Series Convergence2

Use the limit comparison test to see if the series $\frac{1}{\sqrt{n^2 + 2}}$ converges or diverges.

The limit comparison test is a useful tool in determining the convergence or divergence of series, particularly when dealing with series that involve complex expressions. To effectively apply this test, it is essential to identify another series with a known convergence behavior that can be compared to the given series. This often involves recognizing standard benchmark series, such as p-series or geometric series, which have well-established convergence criteria.

In the context of the problem at hand, it is helpful to note the structure of the term $\frac{1}{\sqrt{n^2 + 2}}$, which resembles a p-series of the form $\frac{1}{n^p}$. The challenge lies in identifying an appropriate comparison series, which in this case could be $\frac{1}{n}$, given that the additional constant inside the square root becomes negligible for large $n$.

Once an appropriate comparison series is identified, the limit comparison test involves evaluating the limit of the ratio of the terms from the two series as $n$ approaches infinity. If the result is a finite non-zero number, then both series either converge or diverge together. This strategic approach highlights the importance of understanding series behavior in the broader context of sequence and series convergence, a fundamental topic in calculus.