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The Root Test is a useful tool to determine the convergence characteristics of an infinite series, especially when the terms include exponential functions or powers. If we have a series \( \sum_{n=1}^{\infty} a_n \), the Root Test involves evaluating the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). This test can give us clear insights:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test cannot decide the convergence, and other tests might be needed.

In the given series \( \sum_{n=1}^{\infty} n\left(\frac{4}{5}\right)^{n} \), the application of the Root Test reveals \( \lim_{n \to \infty} \sqrt[n]{n \left(\frac{4}{5}\right)^n} = \frac{4}{5} \). Since \( \frac{4}{5} < 1 \), the series converges absolutely. The Root Test is a powerful method that simplifies the determination of absolute convergence without needing to manipulate series terms.

A geometric series is one in which each term is a constant multiple of the previous one, expressed in the form \( a + ar + ar^2 + ar^3 + \ldots \). The series is defined by its first term \( a \) and common ratio \( r \). Key characteristics of a geometric series include:

• The series converges if the absolute value of the common ratio \( |r| < 1 \).
• The sum of an infinite geometric series can be calculated as \( \frac{a}{1-r} \) when \( |r| < 1 \).
• If \( |r| \geq 1 \), the series diverges.

In our exercise, while the series \( \sum_{n=1}^{\infty} n\left(\frac{4}{5}\right)^n \) resembles a geometric series due to the term \( \left(\frac{4}{5}\right)^n \), the presence of \( n \) complicates it. \( n \) acts as a linearly increasing factor, indicating it's not a pure geometric series, but understanding geometric series concepts helps recognize the growth pattern in the exponents.

Absolute convergence is a stronger form of convergence, ensuring that a series converges even if all its terms are replaced by their absolute values. If a series \( \sum_{n=1}^{\infty} a_n \) is absolutely convergent, the series \( \sum_{n=1}^{\infty} |a_n| \) must also converge. This concept is particularly useful because:

• If a series converges absolutely, it also converges in the usual sense (simple convergence).
• Convergence is not affected by rearranging the terms of an absolutely convergent series.
• The Root Test is one of the tools to determine absolute convergence, as seen in our exercise where \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \) was used.

In the exercise provided, because the Root Test showed a limit of \( \frac{4}{5} \), we identified the series \( \sum_{n=1}^{\infty} n\left(\frac{4}{5}\right)^n \) as absolutely convergent. Understanding absolute convergence is vital as it extends the stability of convergence to many manipulations and theoretical scenarios not guaranteed by simple convergence.