Problem 11 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

The series diverges as \( \lim_{{n \to \infty}} a_n = 1 \neq 0 \).

Step 1: Identify the Divergence Test

The Divergence Test, also known as the n-th term test for divergence, states that if \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) diverges. If \( \lim_{{n \to \infty}} a_n = 0 \), the test is inconclusive.

Step 2: Define the Terms of the Series

The n-th term of the series is \( a_n = \frac{2^n + 1}{2^n + n^2} \). This expression will be used to evaluate the limit as \( n \) approaches infinity.

Step 3: Apply Limit to Series Terms

Evaluate the limit of the n-th term as \( n \) approaches infinity: \( \lim_{{n \to \infty}} \frac{2^n + 1}{2^n + n^2} \).

Step 4: Simplify the Limit Expression

Divide the numerator and the denominator by the dominant term, \( 2^n \), resulting in: \( \lim_{{n \to \infty}} \frac{1 + \frac{1}{2^n}}{1 + \frac{n^2}{2^n}} \).

Step 5: Evaluate the Limit

As \( n \to \infty \), \( \frac{1}{2^n} \to 0 \) and \( \frac{n^2}{2^n} \to 0 \). Therefore, the expression simplifies to \( \lim_{{n \to \infty}} \frac{1+0}{1+0} = 1 \).

Step 6: Conclude Based on the Divergence Test

Since \( \lim_{{n \to \infty}} a_n = 1 eq 0 \), by the Divergence Test, the series \( \sum_{n=1}^{\infty} \frac{2^n + 1}{2^n + n^2} \) diverges.

Key Concepts

Series Convergence

When studying infinite series, one of the initial questions mathematicians ask is whether the series converges or diverges. Convergence in this context means that when you add an infinite number of terms from a series, the total sum approaches a particular number. In simpler terms, a convergence indicates that as you go on adding terms from the series, it closes in on a fixed value. On the other hand, if a series does not settle towards any particular number, it is said to diverge. For example, the series \( \sum_{n=1}^{\infty} \frac{2^n + 1}{2^n + n^2} \) diverges, meaning its sum expands indefinitely without approaching a particular limit. Understanding whether a series converges or diverges is crucial in many areas of mathematics and can be essential for determining the behavior of series in calculus.

Limit Evaluation

Evaluating limits is a fundamental tool used in determining the convergence or divergence of an infinite series. In the context of the divergence test, the focus is on finding the limit of the n-th term as \( n \to \infty \). For the series \( \sum_{n=1}^{\infty} \frac{2^n + 1}{2^n + n^2} \), evaluating the limit of the n-th term involves analyzing the expression \( \lim_{{n \to \infty}} \frac{2^n + 1}{2^n + n^2} \). To do so:

• Identify dominant terms, which simplifies the process of limit evaluation.
• In this example, both the numerator and denominator are dominated by \( 2^n \), making it a powerful element to focus on.
• Divide the expression by \( 2^n \), simplifying to \( \frac{1 + \frac{1}{2^n}}{1 + \frac{n^2}{2^n}} \).

As \( n \to \infty \), \( \frac{1}{2^n} \to 0 \) and \( \frac{n^2}{2^n} \to 0 \), further simplifying the limit result to 1, which plays a key role in deciding the convergence of the series.

Infinite Series

Infinite series are intriguing mathematical structures composed of an endless succession of terms. In many applications, understanding the behavior of these series is essential. An example of an infinite series is \( \sum_{n=1}^{\infty} \frac{2^n + 1}{2^n + n^2} \). Each term in this series contributes variably based on increasing values of \( n \). Even though each term separately may be small or diminishing as \( n \to \infty \), the whole behavior of the series must be considered to see if the sum converges to a specific number or diverges by growing without bound. Infinite series also apply to various practical and theoretical problems, from calculating areas under curves to solving differential equations. They are a central point in calculus education, often requiring a keen eye for detail and a comprehensive understanding of convergence tests.

n-th Term Test

The n-th Term Test, also called the Divergence Test, is the initial step in checking the convergence of a series. It is a basic yet powerful tool. According to this test:

• If \( \lim_{{n \to \infty}} a_n eq 0 \), then \( \sum_{n=1}^{\infty} a_n \) definitely diverges.
• However, if \( \lim_{{n \to \infty}} a_n = 0 \), the result is inconclusive, urging us to employ other tests for further analysis.

In the given exercise, the test successfully points out the nature of the series. For \( \sum_{n=1}^{\infty} \frac{2^n + 1}{2^n + n^2} \), the n-th term test reveals that \( \lim_{{n \to \infty}} \frac{2^n + 1}{2^n + n^2} = 1 \), which is not equal to zero. Therefore, by the n-th term test, we can confidently deduce that this series diverges. This tool is essential for quickly identifying series that do not converge, saving time and effort.