Question

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{n}=n / 2^{n} $$

Short answer

The series converges by the root test because \( \lim_{n \to \infty} \sqrt[n]{|a_n|} < 1 \).

Step-by-step solution

Identify Parameters

First, we examine the given series \( a_n = \frac{n}{2^n} \). We need to determine the convergence of the series \( \sum_{n=1}^{\infty} a_n \) using the root test.

Apply Root Test Formula

The root test involves analyzing the expression \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). In this case, \(|a_n| = \frac{n}{2^n}|\), and we need to compute \( \lim_{n \to \infty} \sqrt[n]{\frac{n}{2^n}} \).

Simplify the Expression

To simplify, crown the formula as \( \frac{1}{2} \cdot \sqrt[n]{n} \) because \( \sqrt[n]{\frac{1}{2^n}} = \left(\frac{1}{2}\right) \). The expression is now \( \lim_{n \to \infty} \frac{1}{2} \cdot \sqrt[n]{n} \).

Evaluate the Limit

Evaluate \( \lim_{n \to \infty} \sqrt[n]{n} \). This equals 1 as the exponential growth of \( n^{\frac{1}{n}} \) approaches 1 when \( n \to \infty \). Therefore, \( \lim_{n \to \infty} \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Determine Convergence

In the root test, if \( \lim_{n \to \infty} \sqrt[n]{|a_n|} < 1 \), the series converges. Since the limit we found is \( \frac{1}{2} \)< 1, we can conclude that the series \( \sum_{n=1}^{\infty} \frac{n}{2^n} \) converges.

Key concepts

Convergence of Series

Convergence of series is a fundamental concept in calculus and mathematical analysis. When evaluating whether a series converges, the primary goal is to determine if the series approaches a specific value as more terms are added. The convergence of a series is a crucial attribute that tells us whether the infinite sum of its terms results in a finite value.

To explore convergence, several tests are available, each suitable for different types of series. Here are some key points about convergence:

• When a series converges, the sum of its infinite terms tends to a single finite number, known as the series' limit.
• If a series diverges, the sum does not settle at any finite value, potentially increasing indefinitely.
• Common methods to test for convergence include the root test, ratio test, and comparison test.
• The root test is particularly useful for series where each term involves powers or roots, allowing us to simplify the analysis of such expressions.

For the series in question, we used the root test, a specific technique to handle expressions involving powers. Mastering convergence can significantly deepen your understanding of infinite series and their behavior.

Infinite Series Analysis

Infinite series analysis involves studying series that extend indefinitely. It's a foundational aspect of calculus where we look at the behavior of series as they approach infinity.

Infinite series take the form \( \sum_{n=1}^{\infty} a_n \) and can describe numerous phenomena in mathematics and the physical world. Here's what you should keep in mind about infinite series analysis:

• Infinite series can depict functions, sequences, or real-world models under certain asymptotic conditions.
• Series are categorized into various types, such as geometric, arithmetic, and power series. Each type offers unique properties and applications.
• Understanding series involves breaking them down into simpler parts and studying their behavior using various mathematical tools.
• By evaluating the convergence using tests like the root test, we determine if the infinite series sum to a particular value.

In our step-by-step solution, we checked whether the infinite series converges using the root test. This approach helped confirm that the series of interest reaches a finite sum.

Limit Evaluation in Calculus

Limit evaluation is an integral part of calculus that helps us understand the behavior of functions and series as variables approach a particular value or infinity.

When analyzing series like \( \sum_{m=1}^{\infty} \frac{n}{2^n} \), we often seek the limits of terms as the series progresses. These insights help determine the convergence of the series. Here are some fundamental aspects of limit evaluation:

• Limits describe the value that a function or sequence approaches as the input progresses towards a particular point.
• They are foundational to defining concepts such as derivatives and integrals in calculus.
• In series convergence tests, computing limits helps ascertain whether the series sums to a finite value.
• The limit of \( \sqrt[n]{n} \) as \( n o \infty \) is a notable example. This expression evaluates to 1, which is a common result used in tests like the root test.

For the given problem, evaluating the limit allowed us to confirm that the infinite series converged as the limit was less than 1. This underscored the utility of limit evaluation in understanding the long-term behavior of series.