91Ó°ÊÓ

Question: Explain the concept of the Limit Comparison Test and its applications in determining the convergence or divergence of two series, using an example to illustrate its application. Answer: The Limit Comparison Test is a method used to determine the convergence or divergence of two series by comparing their growth rates. If the limit of the ratio between the terms of the two series exists and is positive, it indicates that the two series either both converge or both diverge. This is because for large values of n, the terms of both series grow at a similar rate. An example of the Limit Comparison Test application is when comparing the series \(\sum_{n=1}^{\infty} \frac{n}{n^3+1}\) and \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). By calculating the limit of the ratio between their terms and evaluating it, we obtain a limit value of 1, which is positive. Since the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) is a convergent p-series, we can conclude that \(\sum_{n=1}^{\infty} \frac{n}{n^3+1}\) also converges, based on the Limit Comparison Test.