Problem 19 - Essential Calculus Early Transcendentals

Question

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{n}{5^{n}} $$

Step-by-Step Solution

Answer

The series converges absolutely by the Ratio Test.

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{n}{5^{n}} \). This is an example of a power series, where each term takes the form of \( \frac{n}{r^n} \) with \( r = 5 \).

Step 2: Test for Absolute Convergence Using the Ratio Test

The Ratio Test states that for a series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), and if \( L < 1 \), the series converges absolutely.For the series \( a_n = \frac{n}{5^n} \), let's find \( \frac{a_{n+1}}{a_n} \):\[\frac{a_{n+1}}{a_n} = \frac{\frac{n+1}{5^{n+1}}}{\frac{n}{5^n}} = \frac{n+1}{n} \times \frac{1}{5} = \frac{n+1}{5n}.\]Evaluating the limit:\[\lim_{n \to \infty} \frac{n+1}{5n} = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{5} = \frac{1}{5} = 0.2.\]Since \( 0.2 < 1 \), the series converges absolutely.

Step 3: Verify No Need for Conditional Convergence Test

Since the Ratio Test indicates that the series converges absolutely, this means the series does not require testing for conditional convergence. A series that converges absolutely cannot be only conditionally convergent.

Key Concepts

Absolute Convergence

When we talk about absolute convergence, we are interested in whether the series converges when we take the absolute value of each term. A series \( \sum a_n \) is said to be absolutely convergent if the series \( \sum |a_n| \) also converges. Absolute convergence is a stronger form of convergence than regular convergence. If a series is absolutely convergent, it is also convergent in the usual sense. This fact sometimes makes it easier to determine the behavior of a series, especially when dealing with alternating series or series with both positive and negative terms. In the exercise above, the absolute convergence of the series is determined using the Ratio Test, which gives a very reliable method for evaluating this property.

Ratio Test

The Ratio Test is a powerful tool used to ascertain the convergence or divergence of a series. This test works by examining the limit of the ratio of successive terms in a series. Specifically, for a series \( \sum a_n \), the Ratio Test computes:

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \).
• If \( L < 1 \), the series is absolutely convergent.
• If \( L > 1 \) or is infinite, the series is divergent.
• If \( L = 1 \), the test is inconclusive.

In our example with the series \( \sum_{n=1}^{\infty} \frac{n}{5^{n}} \), applying the Ratio Test showed \( L = 0.2 \), which is less than 1. This means that the series converges absolutely. The ease and precision of the Ratio Test makes it a favorite method for dealing with power series and other similar series.

Power Series

Power series are series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) are coefficients and \( x \) is a variable. These series are essential in mathematics as they form the basis of Taylor and Maclaurin series, enabling functions to be expressed as sums of their derivatives at a single point. The series encountered in the exercise, \( \sum_{n=1}^{\infty} \frac{n}{5^{n}} \), can be viewed as a type of geometric series or power series with a diminishing term \( r = \frac{1}{5} \) affecting the convergence. The behavior of power series is highly dependent on the base "r" in each term's denominator. The general convergence properties guarantee that power series can be used to approximate functions very closely within a certain range, known as the interval of convergence.