91Ó°ÊÓ

The Limit Comparison Test is a handy tool for determining the convergence or divergence of a series. It involves comparing the series in question with another series whose convergence behavior is already known. Here’s how it works:

• Select a comparison series, often a simple one like a p-series, that has similar general characteristics as the series you’re examining.
• Calculate the limit of the ratio of the series in question’s terms to the terms of the comparison series as the index goes to infinity.

If this limit is a positive, finite number, the two series will either both converge or both diverge.

In the given problem, we compared \(\sum_{n=1}^{\infty} \frac{1}{n^2 + 4}\) to the p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). Since the limit of their ratio is \(1\), a positive finite number, both series exhibit the same convergence behavior: they both converge.

A p-series is a type of infinite series in the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(p\) is a positive constant. P-series are fundamental in the study of convergence, and knowing when they converge or diverge can be very helpful.

The key takeaway about p-series is:

• If \(p > 1\), the series converges.
• If \(p \leq 1\), the series diverges.

For example, the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) is a p-series with \(p = 2\), and therefore, it converges. In the original solution, this fact was crucial because it served as the reliable basis for comparison in the Limit Comparison Test. By comparing with a known convergent p-series, we confidently establish that the original series also converges.

A positive term series is a series where each term is positive. Understanding the nature of these series is important because they have specific tests for convergence, like the Limit Comparison Test and ratio tests.

Positive term series make it easier to apply these tests since all terms are non-negative, which avoids the issues of alternating signs in terms. Such series ensure stability in convergence behavior when applying various convergence tests. Our given series, \(\sum_{n=1}^{\infty} \frac{1}{n^2+4}\), is an example of a positive term series because each term, \( \frac{1}{n^2+4}\), remains positive for all \(n > 0\).

This property greatly influences how we select which method to apply for determining convergence. In this exercise, the positive terms imply that tests for bounding or comparing with other known-series, such as the Limit Comparison Test, are likely to be effective and reliable.