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The Geometric Series Test is a powerful tool for determining the convergence or divergence of an infinite series. This applies particularly to geometric series of the form \( \sum_{n=1}^{\infty} ar^n \). In this context, \( a \) is the first term, and \( r \) is the common ratio of the series.
The main rule for this test is simple:

• If the absolute value of the common ratio \( |r| \) is less than 1, the series converges.
• If \( |r| \) is greater than or equal to 1, the series diverges.

In the given problem, the series could be approximated as a geometric series \( \left( \frac{5}{4} \right)^n \) for large \( n \). Here, the common ratio \( r = \frac{5}{4} \) is greater than 1, indicating divergence according to the Geometric Series Test. This makes it clear and easy to understand the behavior of the series.

Exponential functions are functions of the form \( a^n \), where \( a \) is a constant and \( n \) represents the exponent. These functions are characterized by their rapid growth or decay, depending on the value of \( a \).
- If \( a > 1 \), the function grows exponentially.- If \( 0 < a < 1 \), the function decays exponentially.In the series \( \sum_{n=1}^{\infty} \frac{5^{n}}{4^{n}+3} \), the numerator \( 5^n \) is an exponential function that grows steadily as \( n \) increases. Comparatively, the term \( 4^n + 3 \) in the denominator can be approximated by \( 4^n \) for large \( n \), since \( 4^n \) dominates \( 3 \).
Understanding the behavior of exponential functions helps in breaking down complex series terms, simplifying comparison with geometric sequences, and applying convergence tests like the Geometric Series Test.

When determining the convergence or divergence of a series, divergence analysis involves examining whether the series grows unbounded. We typically check the limit of the series terms as \( n \to \infty \).
In the given problem, the focus is on terms \( \frac{5^n}{4^n + 3} \) for large \( n \). As \( n \to \infty \), the term \( 5^n \) in the numerator grows faster than \( 4^n \). Evaluating the limit turns the fraction into \( \left( \frac{5}{4} \right)^n \), demonstrating an increase with each term.
The divergence analysis reveals that \( \left( \frac{5}{4} \right)^n \) essentially becomes a base for comparison, guiding us toward a conclusion that the sequence does not converge towards zero. Instead, because the limit of the terms does not equal zero, and the ratio is greater than 1, the series diverges. This analysis plays a crucial role, ensuring that such series behaviors are predicted accurately.