Problem 15 - Thomas Calculus

Question

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{n}{n+1} $$

Step-by-Step Solution

Answer

The series diverges because the sequence term's limit is 1, not 0.

Step 1: Test for Divergence

Start by applying the Divergence Test. For the Divergence Test, if \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum a_n \) diverges. Here, \( a_n = \frac{n}{n+1} \).

Step 2: Calculate the Limit of the Sequence Terms

Calculate \( \lim_{{n \to \infty}} \frac{n}{n+1} \). Simplify the terms by equating it to \( \lim_{{n \to \infty}} \frac{n}{n+1} = \lim_{{n \to \infty}} \frac{1}{1+\frac{1}{n}} \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \), so the limit becomes 1.

Step 3: Determine Series Divergence

Since \( \lim_{{n \to \infty}} \frac{n}{n+1} = 1 eq 0 \), by the Divergence Test, the series \( \sum \frac{n}{n+1} \) diverges.

Key Concepts

Series Convergence

When discussing series, "convergence" refers to the behavior of the sum of the series as the number of terms approaches infinity. In simple terms, a series is said to converge if the sum of its infinite terms approaches a specific finite value. Conversely, a series diverges if the sum does not approach a particular value.

To determine whether a series converges or diverges, various tests can be employed, such as the **Convergence Test**, **Ratio Test**, or the **Divergence Test**. Each test provides a different method to assess the behavior of the series. It's essential to choose the appropriate test for a given series to make accurate determinations.

• A diverging series does not settle at a limit.
• A converging series approaches a finite limit.

In many mathematical problems, correctly identifying whether a series converges or diverges is crucial for further analysis and applications.

Infinite Series

An infinite series is an expression obtained by adding an infinite number of terms together. Such series are often represented as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the general expression for the terms in the sequence.

Infinite series are a cornerstone of calculus and analysis, playing a key role in mathematical fields ranging from analysis to physics. The divergence or convergence of an infinite series is determined by the behavior of its terms as the number of terms increases.

• An infinite series can represent functions, natural phenomena, or theoretical constructs.
• They help in approximating functions and solving differential equations.

Understanding and manipulating infinite series are key skills in advanced mathematics, providing insights into both theoretical and practical problems.

Limit of a Sequence

The concept of a limit of a sequence is fundamental to understanding series and calculus. The limit of a sequence \( \{a_n\} \) occurs when the terms of the sequence approach a specific value \( L \) as \( n \) (the index of the terms) approaches infinity. If the terms approach such a value, we write \( \lim_{n \to \infty} a_n = L \).

Limits are crucial in determining whether a series converges or diverges. For example, in the sequence \( \frac{n}{n+1} \), calculating the limit as \( n \to \infty \) allows us to apply the **Divergence Test** effectively.

• If \( \lim_{n \to \infty} a_n = 0 \), it does not automatically imply convergence of the series.
• If \( \lim_{n \to \infty} a_n eq 0 \), as in this exercise where the limit is 1, the series diverges.

Understanding the behavior of limits provides deeper insights into how sequences and series behave over infinite lengths.

Calculus

Calculus is a branch of mathematics that studies change and motion, building the foundation for much of modern mathematical analysis. It encompasses various concepts, such as derivatives, integrals, limits, and infinite series, providing the tools necessary for modeling and solving real-world problems.

In the context of series and sequences, calculus helps in understanding and applying different tests for convergence or divergence. The **Divergence Test** used in this exercise is a classic example of applying calculus principles.

• Calculus provides methods for calculating the behavior of changing quantities.
• It offers the framework for solving complex problems in engineering and the physical sciences.

Mastering calculus concepts, such as series, limits, and continuity, equips students with the knowledge to tackle advanced mathematical challenges, revealing the intricate beauty of mathematics.