Question

Use the Limit Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \sin (1 / n)$$

Short answer

The series diverges; it is compared to the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \).

Step-by-step solution

Identify the series

The given series is \( \sum_{n=1}^{\infty} \sin(1/n) \). To apply the Limit Comparison Test, we must compare this series to a known convergent or divergent series.

Choose a comparison series

Notice that as \( n \to \infty \), \( \sin(1/n) \approx 1/n \). A natural choice for comparison is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known to be divergent.

Compute the limit for the Limit Comparison Test

Calculate \( \lim_{n \to \infty} \frac{\sin(1/n)}{1/n} \). This is equivalent to \( \lim_{n \to \infty} n \sin(1/n) = \lim_{n \to \infty} \frac{\sin(1/n)}{1/n} \).

Simplify the limit

Recognize that \( \frac{\sin(x)}{x} \to 1 \) as \( x \to 0 \). So, \( \lim_{n \to \infty} \frac{\sin(1/n)}{1/n} = 1 \).

Apply the Limit Comparison Test conclusion

Since the limit is a finite positive number (1), by the Limit Comparison Test, \( \sum_{n=1}^{\infty} \sin(1/n) \) and \( \sum_{n=1}^{\infty} \frac{1}{n} \) have the same convergence behavior. Since the harmonic series diverges, the given series also diverges.

Key concepts

Convergence

In mathematics, convergence is a fundamental concept used to describe the behavior of infinite series or sequences as they approach a specific limit. An infinite series \( \sum a_n \) is said to converge if its sequence of partial sums \( S_n = a_1 + a_2 + ... + a_n \) tends to a definite value as \( n \to \infty \). Conversely, if the partial sums do not approach any limit, the series is termed divergent.
Understanding convergence is crucial when analyzing series, as it helps determine whether the sum of an infinite set of terms results in a finite number. Several tests exist for checking convergence, one of which is the Limit Comparison Test used in the original exercise.

• A series can converge conditionally or absolutely.
• Absolute convergence means that the series formed by taking the absolute values of the terms also converges.
• Conditional convergence implies convergence without the guarantee that the series of absolute values will converge.

Recognizing convergence enables further exploration in calculus and analysis, offering meaningful insights into infinite processes.

Harmonic Series

The harmonic series is one of the most notable examples in the study of infinite series given by \( \sum_{n=1}^{\infty} \frac{1}{n} \). Despite its simple form, it is a classic example of a divergent series.
The harmonic series reflects the growth of partial sums when each term added is the reciprocal of a positive integer. It's essential in many fields of mathematics, particularly in analysis and number theory.

• The divergence of the harmonic series is a surprising result since each term becomes infinitesimally small as \( n \to \infty \).
• It serves as a benchmark for testing the nature of series when using the Limit Comparison Test.
• When compared with similar series by rate and behavior, it provides valuable understanding for determining convergence.

The harmonic series, despite diverging, is foundational and appears in various mathematical contexts.

Divergent Series

A divergent series is an infinite series that does not converge to a finite limit. For a series \( \sum a_n \), divergence indicates that its sequence of partial sums does not approach a finite value as \( n \to \infty \).
This concept is broad, covering any series where terms do not accumulate to a specific total under repeated addition.

• Series that grow unboundedly, oscillate, or exhibit erratic behavior typically fall into this category.
• Divergence does not imply randomness but rather a lack of finite total.
• Recognizing divergence is crucial before attempting to use a series in practical applications like an evaluation of integrals or real-world modeling.

In the exercise, recognizing divergence through the harmonic series' properties helps conclude the behavior of the given series using the Limit Comparison Test.

Calculus

Calculus is a branch of mathematics with primary focus on change and motion, utilizing techniques and theories to study rates of change (differential calculus) and accumulation of quantities (integral calculus). When working with infinite series, calculus provides powerful tools to determine convergence or divergence.
Series and sequences are components of calculus, allowing mathematicians to approximate and solve complex problems.

• The concept of limits, pivotal in calculus, underpins the evaluation of infinite series and sequence behavior.
• Differential calculus highlights how sequences can approximate functions, much like Taylor and Maclaurin series.
• Integral calculus is often used to compute areas under curves, a concept related to summing infinite series.

Calulus not only expands the tools available for rigorous mathematical analysis but also bridges the gap between finite calculations and infinite investigations.

Infinite Series

An infinite series is the summation of an infinite sequence of terms. Represented as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) are real or complex numbers, their study involves testing for convergence or divergence.
Infinite series appear in various mathematical disciplines, serving as foundational structures for approximations, function representations, and solutions to differential equations.

• Understanding each term’s effect on the series as a whole helps identify convergence patterns.
• Infinite series are used in algorithms, signal processing, and the expanse of calculus itself.
• Key tests, like the Limit Comparison Test, provide a straightforward method to analyze them.
  They relate terms of unfamiliar series to known series to infer behavior.

By grasping infinite series, one gains comprehension over a wide array of problems that involve continuous processes or approximations.