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The Comparison Test is a technique to determine the convergence of a series by comparing it to another series whose behavior is known. To apply this test, we consider a series \( \sum a_n \) and compare it with another series \( \sum b_n \), where all terms \( a_n \) and \( b_n \) are positive, for large \( n \).

• If \( a_n \leq b_n \) for all \( n \) and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
• If \( a_n \geq b_n \) for all \( n \) and \( \sum b_n \) diverges, then \( \sum a_n \) also diverges.

When using the Comparison Test, you must choose a suitable \( \sum b_n \) that simplifies the comparison. Often, \( b_n \) is selected based on the dominant term of \( a_n \) for large \( n \), making it easier to evaluate the behavior of the original series.

The Limit Comparison Test is a useful tool for analyzing series, especially when the Comparison Test might be challenging to apply. With this test, we compare two series \( \sum a_n \) and \( \sum b_n \) by examining the limit of their term ratios as \( n \) approaches infinity.

The Limit Comparison Test is applied as follows:

• Calculate \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If \( 0 < L < \infty \), then both series \( \sum a_n \) and \( \sum b_n \) either converge or diverge together.

This test is powerful because it allows you to use a series with known convergence behavior to test another. It's especially handy when choosing \( b_n \) as a simpler form of \( a_n \) by focusing on their dominant terms for large \( n \).

A p-series is a specific type of series expressed in the form \( \sum \frac{1}{n^p} \), where \( p \) is a positive constant. The convergence or divergence of a p-series is determined by the value of \( p \), making it a handy reference when dealing with series convergence.

A p-series converges if \( p > 1 \) and diverges if \( p \leq 1 \). This makes it easy to identify the behavior of many series quickly, by comparing them to a known p-series. In the given exercise, comparing the series \( \sum_{n=4}^{\infty} \frac{\sqrt{n}}{n^2-3} \) to the p-series \( \sum \frac{1}{n^{3/2}} \), where \( p = \frac{3}{2} \), allows us to predict convergence, since \( \frac{3}{2} > 1 \).

Recognizing a series as a p-series helps simplify the process of checking convergence by applying straightforward rules, without complicated calculations.

The Integral Test is another method for determining the convergence of series. It is based on the relationship between a series and an improper integral. The test can be performed on series \( \sum a_n \) where the functions \( f(n) = a_n \) are continuous, positive, and decreasing for large \( n \).

The test involves the following steps:

• Evaluate the improper integral \( \int_{N}^{\infty} f(x) \, dx \), choosing an appropriate \( N \).
• If the integral converges, then the series \( \sum a_n \) also converges.
• If the integral diverges, then the series \( \sum a_n \) also diverges.

The Integral Test is particularly useful when dealing with series that involve more complex forms for which geometric or p-series comparisons aren’t as clear. It provides a bridge between series convergence and the well-studied behavior of improper integrals, allowing for a different angle of approach.