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The Comparison Test is a powerful tool to determine the convergence or divergence of a series. It works by comparing a series you're interested in with another series whose behavior (converging or diverging) is already known. To use the Comparison Test, identify a series whose terms are similar to or "simplify" your series' terms. You then determine if these terms are generally less than or equal to the comparison series' terms or greater. - If the terms of your series are less than or equal to the terms of a converging series, your series also converges. - Conversely, if they are greater than the terms of a diverging series, your series diverges.In the given problem, we found:- The series \(\sum_{n=1}^{\infty} \left(\frac{2}{3}\right)^n\) is geometric with a known convergence, as its common ratio is less than one.- We approximated \(\frac{n+4^n}{n+6^n} \approx \left(\frac{2}{3}\right)^n\) for large \(n\), ensuring it fulfills the necessary conditions of the Comparison Test.

A geometric series is a series where each term is found by multiplying the previous term by a constant called the common ratio, denoted as \( r \). It can be written as \( a + ar + ar^2 + ar^3 + \cdots \), or in summation notation: \(\sum_{n=0}^{\infty} ar^n\).Key properties of a geometric series include:- The series converges when the absolute value of \( r \) is less than 1 (\(|r| < 1\)).- It diverges when \(|r| \geq 1\).The geometric series used in this problem is \(\sum_{n=1}^{\infty} \left(\frac{2}{3}\right)^n\).This series has \( r = \frac{2}{3} \), which is less than 1, so it converges. Such series are instrumental in the Comparison Test because their convergence properties are well-established.

Understanding dominant terms is essential for analyzing polynomials or expressions within limits, such as in series. As \( n \) becomes extremely large, some terms influence the value more than others.The key to identifying dominant terms is to focus on comparing rates of growth:- Exponential terms (e.g.,\(4^n\),\(6^n\)) grow much faster than polynomial terms (e.g., \(n\)).- Therefore, in the expression \(\frac{n+4^n}{n+6^n}\), the \(4^n\) in the numerator and \(6^n\) in the denominator are the dominant terms for large \(n\).By recognizing \(\frac{4^n}{6^n}\), simplifying to \(\left(\frac{2}{3}\right)^n\), we efficiently work within the Comparison Test framework. Evaluating dominant terms allows a quick grasp of a series' long-term behavior.