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Understanding when a series converges or diverges is essential in calculus. A series $$\sum_{n=1}^{\infty} a_n$$converges if the sum of its infinite terms approaches a specific limit as more terms are added. Otherwise, it diverges. To assess convergence, mathematicians often employ various tests and comparisons.

• Convergence: The series equals a finite value as you add more terms.
• Divergence: The series does not settle down to a single number.

These concepts are foundational when working with infinite sequences and are essential for understanding series behavior in mathematics. A series that diverges often grows indefinitely or oscillates without nearing a single unchanging value.

A geometric series is a series with a constant ratio between successive terms. It takes the form$$\sum_{n=0}^{\infty} ar^n,$$where \(a\) is the initial term and \(r\) is the ratio. The convergence of a geometric series depends on the absolute value of \(r\).

• The series converges if \(|r| < 1\).
• The series diverges if \(|r| \geq 1\).

In our example, the original series contains a term \(\left(\frac{5}{4}\right)^n\), which is geometric with a ratio greater than one \(\left(\frac{5}{4} > 1\right)\), leading to divergence. Therefore, recognizing geometric series elements helps quickly determine properties like convergence or divergence.

Simplifying a series expression makes it easier to analyze and compare. The original series $$\sum_{n=1}^{\infty} \frac{5^n}{\sqrt{n} 4^n}$$can be expressed in a simpler form. By rewriting the expression as $$\frac{5^n}{\sqrt{n} \cdot 4^n} = \left(\frac{5}{4}\right)^n \cdot \frac{1}{\sqrt{n}},$$we separate the geometric component from the variable component.

• Identify dominance: Which part of the series affects behavior most.
• Simplification leads: Break down into understandable components.

This transformation is crucial for applying methods like the Limit Comparison Test since it clarifies how different parts of the series behave as \(n\) increases.

When using the Limit Comparison Test, calculating the limit of the ratio between the given series and a comparison series helps determine whether both share the same convergence behavior. For the series $$\sum_{n=1}^{\infty} \frac{5^n}{\sqrt{n} 4^n},$$we considered a geometric series \(\sum_{n=1}^{\infty} \left(\frac{5}{4}\right)^n\). By evaluating the limit$$\lim_{n \to \infty} \frac{\left(\frac{5}{4}\right)^n \cdot \frac{1}{\sqrt{n}}}{\left(\frac{5}{4}\right)^n} = \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0,$$we see that divides out the main geometric series part.

• Limit Interpretation: Determines the behavior of the series as \(n\) approaches infinity.
• Result Application: The series follows the dominant component.

Even when the limit evaluates to 0, the divergence of the comparison series suggests the original also diverges due to the overpowering exponential growth in both.