Problem 69 - Essential Calculus

Question

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n}{4^{n}} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{n}{4^{n}} \) is convergent according to the Root Test.

Step 1: Identifying the terms of the series

In the given series \( \sum_{n=1}^{\infty} \frac{n}{4^{n}} \), the nth term is \(a_n = \frac{n}{4^n}\).

Step 2: Applying the Root Test

In the Root Test, calculate the limit as n approaches infinity of the nth root of the absolute value of the nth term, i.e., \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). Here, \( a_n = \frac{n}{4^n} \) so \( \sqrt[n]{|a_n|} = \sqrt[n]{\frac{n}{4^n}} = \left( \frac{n}{4^n} \right)^{1/n} = \frac{n^{1/n}}{4} = \frac{1}{4}\cdot n^{1/n} \).

Step 3: Calculating the limit

Next, calculate \( \lim_{n \to \infty} \frac{1}{4}\cdot n^{1/n} \). As n approaches infinity, \( n^{1/n} \) approaches 1, so the limit is \( \frac{1}{4} \).

Step 4: Determining the convergence

Since the limit \( L=\frac{1}{4} < 1 \), according to the Root Test, the series \( \sum_{n=1}^{\infty} \frac{n}{4^{n}} \) is convergent.

Key Concepts

Convergence

Convergence in infinite series refers to whether the series approaches a specific number or value as more terms are added. When we say a series converges, it means that adding infinitely many terms will result in a finite sum.
To check for convergence, different tests are used, one of which is the Root Test. Convergence is confirmed when the limiting behavior of the series' terms meet the criteria of the convergence tests used.
In the original exercise, the Root Test is applied to determine if the series converges. The series is ultimately found to converge since the limit in the Root Test is less than 1.

Divergence

Divergence in an infinite series means that the sum of the series does not approach any finite number, no matter how many terms you add.
A divergent series can either grow without bound or oscillate indefinitely without settling down to a sum.
When applying tests for convergence, if the result does not meet the criteria, the series is divergent.
In the given exercise, if the result of the Root Test had been greater than or equal to 1, the series would have been divergent.

Infinite series

An infinite series is a sum of infinitely many terms. It's like adding up numbers that don't stop. Each term of an infinite series contributes to finding the overall sum.
In mathematics, we often denote this kind of series with the Greek letter sigma (\( \Sigma \)) followed by a formula that represents the terms.
The original exercise includes such a series: \( \sum_{n=1}^{\infty} \frac{n}{4^n} \). This series has infinitely many terms, each given by the formula \( \frac{n}{4^n} \).
Examining whether such an infinite series converges or diverges is crucial for understanding its behavior and the outcomes it may produce.

Limit calculations

Limit calculations are essential when working with convergence and divergence, especially in applying tests like the Root Test.
The idea is to find the behavior of a sequence or series as the terms get very large, typically when \(n\) approaches infinity.
For the Root Test, we calculate the limit of the nth root of the absolute value of the terms in the series.

• In the example, the nth root calculation gives \( \left( \frac{n}{4^n} \right)^{1/n} \), which simplifies to \( \frac{1}{4} \cdot n^{1/n} \).
• The limit evaluation involves considering what happens to \( n^{1/n} \) as \(n\) becomes very large.

Knowing these limit calculations help us decide if an infinite series will converge to a finite value, like in our given exercise where the limit is found to be \( \frac{1}{4} \), indicating convergence.