Problem 19 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

The Divergence Test is inconclusive for \( \sum_{n=1}^{\infty} \sin(1/n) \).

Step 1: Recognize the Divergence Test

The Divergence Test, also known as the nth-term test for divergence, states: If \( \lim_{{n \to \infty}} a_n eq 0 \), or the limit does not exist, then the series \( \sum_{n=1}^{\infty} a_n \) diverges. However, if \( \lim_{{n \to \infty}} a_n = 0 \), the test is inconclusive.

Step 2: Identify the Sequence Term

Identify the term of the sequence you are testing for divergence: \( a_n = \sin(1/n) \).

Step 3: Evaluate the Limit of the Sequence Term

Calculate \( \lim_{{n \to \infty}} \sin(1/n) \). As \( n \to \infty \), \( 1/n \to 0 \). Since \( \sin(x) \to x \) as \( x \to 0 \), it follows that \( \sin(1/n) \to 0 \). Thus, \( \lim_{{n \to \infty}} \sin(1/n) = 0 \).

Step 4: Conclusion Using the Divergence Test

Based on the result from Step 3, since \( \lim_{{n \to \infty}} \sin(1/n) = 0 \), the Divergence Test is inconclusive. It does not provide any information about the convergence or divergence of the series \( \sum_{n=1}^{\infty} \sin(1/n) \).

Key Concepts

Series Convergence

In mathematics, especially in calculus, understanding whether a series converges or diverges is vital. A series is essentially a sum of terms in a sequence. When we ask if a series converges, we're questioning whether the sum of its infinite terms arrives at a finite number. If it does, the series is convergent; if not, it is divergent.

The Divergence Test, or the nth-term test, is one of the simplest methods to check for divergence. It states that if the limit of the sequence terms is not zero, the series must diverge. However, this test only helps to identify divergence and can’t confirm convergence. If the limit is zero, as in the case of our exercise, where \( \lim_{{n \to \infty}} \sin(1/n) = 0 \), it simply means the Divergence Test is inconclusive. So, additional tests or methods need to be employed to check for convergence.

Limit Evaluation

Evaluating limits is a key skill when working with sequences and series. Limits help us understand the behavior of a sequence as it approaches infinity. When evaluating \( \lim_{{n \to \infty}} \sin(1/n) \), it's constructive to note the behavior of \( \sin(x) \) as \( x \to 0 \).

Because \( \sin(x) \to x \) when \( x \to 0 \), and \( 1/n \to 0 \) as \( n \to \infty \), it follows that \( \sin(1/n) \to 0 \). This evaluation confirms that the limit of the sequence term \( a_n = \sin(1/n) \) is zero. This limit informs us that the Divergence Test is not helpful in this scenario. To fully determine whether a series converges, further analysis beyond the Divergence Test, such as using the Comparison Test or the Integral Test, might be essential.

Sequence Term Identification

Identifying the core term in a sequence is like discovering the building block of a series. For our particular exercise, the term \( a_n = \sin(1/n) \) must be identified and analyzed. Knowing this term allows us to evaluate its behavior over large values of \( n \).

The sequence of interest, \( \{\sin(1/n)\} \), involves applying the formula of the sine function to a reciprocal term. Understanding how the term changes informs us about the potential convergence or divergence of the series. This term must be processed through relevant tests to reach conclusions about the series. Observing that \( \sin(1/n) \to 0 \) gives us limited information initially but is crucial for initial assessments in calculus.