Problem 4 - Calculus Single Variable

Question

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{3^{n}}{3^{n}+4} $$

Step-by-Step Solution

Answer

The series diverges by the Divergence Test.

Step 1: Understand the Divergence Test

The Divergence Test states that for an infinite series \( \sum_{n=1}^{\infty} a_n \), if \( \lim_{n \to \infty} a_n eq 0 \), then the series diverges. However, if \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive and does not provide information about convergence.

Step 2: Identify the Term of the Series

The given series is \( \sum_{n=1}^{\infty} \frac{3^{n}}{3^{n}+4} \). The term of the series is \( a_n = \frac{3^n}{3^n + 4} \). We will use this to apply the Divergence Test.

Step 3: Compute the Limit of the Terms

Calculate \( \lim_{n \to \infty} a_n \).\[\lim_{n \to \infty} \frac{3^n}{3^n + 4} = \lim_{n \to \infty} \frac{3^n}{3^n (1 + \frac{4}{3^n})} = \lim_{n \to \infty} \frac{1}{1 + \frac{4}{3^n}} = \frac{1}{1 + 0} = 1\]

Step 4: Apply the Divergence Test

Since \( \lim_{n \to \infty} a_n = 1 eq 0 \), according to the Divergence Test, the series \( \sum_{n=1}^{\infty} \frac{3^n}{3^n + 4} \) diverges.

Key Concepts

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Think of it as adding up numbers forever. Mathematically, it is expressed as \( \sum_{n=1}^{\infty} a_n \). Here, \( \sum \) denotes a sum, and \( n \) starts from 1 and goes to infinity.

• Each individual term in the sequence is represented by \( a_n \).
• An infinite series may converge to a specific value or diverge, meaning it doesn't settle at any particular value.

Convergence or divergence of a series provides us with important information. It helps us understand whether summing infinitely many terms will result in a finite number or not. This is a key concept when studying calculus and mathematical analysis.

Convergence and Divergence

Convergence and divergence are fundamental properties of an infinite series. When we talk about the convergence of a series, it means that as we add more and more terms, the sum approaches a certain finite value. If a series does not converge, it diverges. This means either the series grows indefinitely or oscillates without approaching a specific value.

• Convergent Series: The sum stabilizes to a finite number.
• Divergent Series: The sum keeps increasing or decreases without limit, or fluctuates endlessly.

To determine if a series converges or diverges, various tests are used. The Divergence Test is one such test; though it directly identifies divergence, it does not confirm convergence. This test checks the limit of the sequence terms \( a_n \). If this limit is not zero, the series diverges.

Limits in Calculus

In calculus, the concept of a limit helps us understand the behavior of functions as they approach a specific point or infinity. It is a foundational idea that supports the definition of derivatives, integrals, and infinite series.When analyzing an infinite series, calculating the limit of its terms \( a_n \) as \( n \to \infty \) tells us something crucial:

• If the limit \( \lim_{n \to \infty} a_n \) equals zero, the series might still converge, but not necessarily.
• If this limit is not zero, the series definitely diverges, as supported by the Divergence Test.

Understanding limits helps us perform more accurate analyses in calculus, allowing us to predict how series and functions behave at extreme values or conditions.