Problem 5 - Calculus Single Variable

Question

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{1}{1+1 / n} $$

Step-by-Step Solution

Answer

The series diverges by the Divergence Test because \( \lim_{n\to\infty} a_n = 1 \neq 0 \).

Step 1: Analyze the terms of the series

The series given is \( \sum_{n=1}^{\infty} \frac{1}{1+1/n} \). Let's simplify the general term \( a_n = \frac{1}{1+1/n} \). By converting to a common denominator, we have: \[ a_n = \frac{1}{1 + 1/n} = \frac{1}{\frac{n+1}{n}} = \frac{n}{n+1}. \]

Step 2: Evaluate the limit of the terms

To apply the Divergence Test, we compute the limit \( \lim_{n\to\infty} a_n \). This becomes: \[ \lim_{n\to\infty} \frac{n}{n+1} = \lim_{n\to\infty} \frac{n}{n+1} = \lim_{n\to\infty} \frac{1}{1 + 1/n}. \] As \( n \to \infty \), \( 1/n \to 0 \), so \[ \frac{1}{1+1/n} \to \frac{1}{1} = 1. \]

Step 3: Apply the Divergence Test

According to the Divergence Test, if \( \lim_{n\to\infty} a_n eq 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) diverges. Since \( \lim_{n\to\infty} a_n = 1 eq 0 \), the series does not converge.

Key Concepts

Infinite Series

An infinite series is essentially a summation of an endless list of numbers. It is represented mathematically as \( \sum_{n=1}^{\infty} a_n \), indicating that the series starts with the first term \( a_1 \) and continues indefinitely. The formation of such a series stems from adding terms in sequence forever.
A classic example is the series of fractions or geometric progressions. Infinite series can either approach some finite value (converge) or grow indefinitely larger (diverge). This distinction is crucial when analyzing their behavior.
Understanding infinite series is fundamental to various advanced mathematics fields, including calculus and analysis. The concept plays a significant role in evaluating functions, especially when determining convergence or divergence.

Limit Evaluation

When dealing with infinite series, evaluating the limit of terms is a key part of understanding their behavior. The limit of a series refers to the value the terms approach as the sequence goes to infinity. In calculus, we often use the notation \( \lim_{n\to\infty} a_n \) to describe this process.
Consider the series in the original exercise: \( a_n = \frac{n}{n+1} \). As \( n \to \infty \), each term approximates closer and closer to 1. Thus, the limit is 1, not zero.

• If the limit of the terms as \( n \to \infty \) is zero, it doesn't necessarily mean the series converges, but it is a good indicator to check further.
• If the limit is not zero, the series cannot converge according to the Divergence Test.

Evaluating limits is important since it helps in deciding whether a series converges or diverges.

Series Convergence

Convergence of a series indicates that the sum of its infinite terms approaches a specific value. For a series \( \sum_{n=1}^{\infty} a_n \) to converge, the limit of its individual terms as \( n \to \infty \) must tend towards zero.
In the context of the Divergence Test, convergence is checked by examining if the limit of the sequence of terms is zero. This test is a preliminary step in analyzing series and is quite straightforward:

• If \( \lim_{n\to\infty} a_n = 0 \), further tests are required to establish convergence.
• If \( \lim_{n\to\infty} a_n eq 0 \), the series is said to diverge.

Hence, convergence concepts enable us to determine the summability of infinite series.

Calculus Problem Solving

Solving calculus problems often involves understanding and applying different tests and concepts around limits, infinite series, and convergence.
The Divergence Test, applied in this exercise, is a straightforward method in calculus to determine if an infinite series diverges based on the behavior of the terms as \( n \to \infty \). However, this is only one tool among many, such as the Ratio Test, Integral Test, and Comparison Test, that can be used depending on the series in question.

• Approach problems by breaking them down: simplify terms.
• Evaluate limits carefully to understand term behavior.
• Apply appropriate tests to assess convergence or divergence.

These steps are crucial for effectively solving complex calculus problems. Mastery of these concepts enables students to tackle a wide range of mathematical challenges more confidently.