Problem 20 - Calculus Single Variable

Question

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty}(n /(n+1)-1 /(n+2)) $$

Step-by-Step Solution

Answer

The series diverges according to the Divergence Test, since the limit of its terms is 1, not 0.

Step 1: Identify the Form of the Series

The given series is \( \sum_{n=1}^{\infty} \left( \frac{n}{n+1} - \frac{1}{n+2} \right) \). This is an infinite series where each term is the difference of two fractions.

Step 2: Simplify Each Term

Find a common denominator for the terms within the parentheses to simplify them. The term \( \frac{n}{n+1} - \frac{1}{n+2} \) can be rewritten as \( \frac{n(n+2) - (n+1)}{(n+1)(n+2)} \). Simplify this expression.

Step 3: Evaluate the Simplified Expression

First, expand the numerator: \( n(n+2) - (n+1) = n^2 + 2n - n - 1 \), which simplifies to \( n^2 + n - 1 \). So, each term becomes \( \frac{n^2 + n - 1}{(n+1)(n+2)} \).

Step 4: Apply the Divergence Test

Calculate \( \lim_{n \to \infty} a_n \) where \( a_n = \frac{n^2 + n - 1}{(n+1)(n+2)} \). Divide the numerator and the denominator by \( n^2 \) to simplify the limit calculation. This gives \( \lim_{n \to \infty} \frac{1 + \frac{1}{n} - \frac{1}{n^2}}{1 + \frac{3}{n} + \frac{2}{n^2}} = 1 \).

Step 5: Interpret the Divergence Test Result

Since \( \lim_{n \to \infty} a_n = 1 eq 0 \), according to the Divergence Test, the series diverges. The Divergence Test states that if the limit of the terms of a series is not zero, the series cannot converge.

Key Concepts

Understanding Infinite Series

When we talk about an infinite series, we're referring to the sum of an endless list of numbers. In mathematical terms, it's expressed as \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) is a term of the series.
This type of series goes on without stopping, reaching towards infinity. The series given in the exercise is infinite because it starts at 1 and keeps going.
Infinite series are important in calculus and many mathematical applications, as they help describe and solve problems involving continuous processes.

• Infinite Nature: The sum never really ends — it tries to reach a specific value.
• Application: Used in real-world calculations, such as physics and engineering problems.

Understanding and manipulating infinite series is crucial to determining whether they have a finite sum, known as convergence, or not, known as divergence.

Simplifying Fractions in the Series

Simplifying fractions is essential when working with formulas or series. The goal is to make the expression easier to manage, especially when you have to perform calculations.
In the step-by-step solution, we began with \( \frac{n}{n+1} - \frac{1}{n+2} \). To simplify, you need a common denominator, which can be daunting but is necessary for simplification.
Here’s what happens:

• Find a common denominator: This was \((n+1)(n+2)\)
• Rewrite each part: Make both parts of the expression share that common denominator.
• Simplify the numerator: Expand and combine like terms to get \( n^2 + n - 1 \).

Each step is important to make the terms manageable, preparing the expression for the limit calculation, which determines the series' behavior as it approaches infinity.

Calculating Limits

Limits help us understand the behavior of functions or sequences as they approach a specific value or infinity.
In this problem, we need to find the limit of \( a_n \) as \( n \rightarrow \infty \) to apply the Divergence Test. The tricky part is simplifying the fractions to make limit calculation straightforward.

• Simplified Expression: Start with \( \frac{n^2 + n - 1}{(n+1)(n+2)} \).
• Dividing by \( n^2 \): reduces both the numerator and denominator to manage the degree of the polynomial.
• Final Limit: After simplification, compute \( \lim_{n \to \infty} \frac{1 + \frac{1}{n} - \frac{1}{n^2}}{1 + \frac{3}{n} + \frac{2}{n^2}} = 1 \).

This step is crucial to see whether the sequence converges or diverges. In this case, since the limit is 1, we proceed to interpretation.

Series Convergence and the Divergence Test

Series convergence is about whether the series adds up to a specific, finite value. If an infinite series converges, its terms approach zero, allowing the series to settle at a certain value.
But not all series converge; some diverge, meaning they never stabilize to a single value. This is where the Divergence Test becomes handy.

• Divergence Test: If \( \lim_{n \to \infty} a_n \, eq \, 0 \), the series diverges.
• In Our Case: We calculated \( \lim_{n \to \infty} a_n = 1 \).
• Conclusion: The series diverges because the limit is not zero.

Understanding convergence is vital in identifying whether or not a series will settle at a specific number, which is essential in fields like analysis and applied mathematics.