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A geometric series is one of the simplest types of series to understand. It's formed by a sequence of terms, each of which is obtained by multiplying the previous term by a constant called the common ratio. In general, a geometric series can be expressed as:\[ S = a + ar + ar^2 + ar^3 + \cdots \]where \( a \) is the first term and \( r \) is the common ratio. An important characteristic of geometric series is that they have a clear rule for determining convergence. If the absolute value of the common ratio \( |r| < 1 \), the series converges. If \( |r| \geq 1 \), the series diverges.

In the exercise, the term \( \left( \frac{3}{4} \right)^n \) represents a geometric series with a common ratio \( r = \frac{3}{4} \), which is less than 1. Therefore, this reference series converges.

The Limit Comparison Test is a useful method for determining the convergence or divergence of a series by comparing it to another series whose behavior is known. This test uses a limit to compare the terms of two series.

Here's how the Limit Comparison Test works:

• Given two series \( \sum a_n \) and \( \sum b_n \) with positive terms, compute the limit \( \lim_{n o \infty} \frac{a_n}{b_n} \).
• If the limit is a positive finite number, both series either converge or diverge together.

In the solution provided, the given series terms \( a_n = \frac{2^n + 3^n}{3^n + 4^n} \) were compared to the geometric series \( b_n = \left( \frac{3}{4} \right)^n \). The limit turned out to be 1.

Since \( \sum b_n \) converges, \( \sum a_n \) converges as well by the Limit Comparison Test.

When solving series problems, especially those involving exponential functions, it's useful to identify dominant terms. A dominant term is the part of an expression that grows the fastest as the index goes to infinity.

In the given series, the terms \( 3^n \) and \( 4^n \) play crucial roles.

• In the numerator, \( 3^n \) eventually surpasses \( 2^n \) as \( n \) becomes very large, because 3 is greater than 2.
• In the denominator, \( 4^n \) dominates \( 3^n \) for the same reasoning, since 4 is greater than 3.

This makes it easier to approximate the series terms by focusing on these dominant terms. The approximation \( \frac{3^n}{4^n} = \left( \frac{3}{4} \right)^n \) simplifies the analysis because it captures the essential behavior of the series as \( n \to \infty \). Recognizing dominant terms helps in correctly applying tests like the Limit Comparison Test for concluding convergence.