Problem 41 - Essential Calculus Early Transcendentals

Question

If \(\Sigma a_{n}\) is a convergent series with positive terms, is it true that \(\sum \sin \left(a_{n}\right)\) is also convergent?

Step-by-Step Solution

Answer

No, \( \sum \sin(a_{n}) \) is not necessarily convergent.

Step 1: Analyze the Given Series

We are given that \( \Sigma a_{n} \) is a convergent series with positive terms. A convergent series implies that the terms \( a_n \) tend to 0 as \( n \to \infty \).

Step 2: Examine Sine Function Properties

Consider the function \( \sin(x) \). For small positive \( x \), \( \sin(x) \) can be approximated by \( x \) because the function \( \sin(x) \) is approximately linear near zero. This means \( \sin(a_n) \approx a_n \) for small positive \( a_n \).

Step 3: Assess Convergence of \( \sum \sin(a_n) \)

Since \( \sin(a_n) \approx a_n \) for small values and \( \sum a_n \) is a convergent series, we initially hypothesize that \( \sum \sin(a_n) \) might also converge. However, convergence of \( \sum \sin(a_n) \) requires more than just \( \sin(a_n) \approx a_n \).

Step 4: Convergence Test

To determine convergence, we need to compare \( \sum \sin(a_n) \) to a known convergent or divergent series. Note that since \( a_n \to 0 \) as \( n \to \infty \), for large \( n \), \( \sin(a_n) \approx a_n \). However, if \( a_n \)s are specifically chosen such that they go to zero rapidly, the series might behave differently. General tests or further examination is needed.

Step 5: Conclusion with Counterexample

Consider a counterexample: let \( a_n = 1/n \). The harmonic series \( \sum 1/n \) diverges; however, \( \sin(1/n) \approx 1/n \) for large \( n \). Convergence cannot be guaranteed, showing that \( \sum \sin(a_n) \) is not necessarily convergent just because \( \Sigma a_{n} \) is.

Key Concepts

Series Convergence Tests

When we talk about series convergence, we're analyzing whether the sum of infinitely many terms approaches a finite number. Major convergence tests help us determine the behavior of a series.

• n-th Term Test: If the terms of a series do not approach zero, the series cannot converge. For a convergent series, each term decreases to zero as the series grows.
• Comparison Test: Compares the series to another series. If you have a series of smaller terms against a known convergent series, then your series converges too. Conversely, if your series has larger terms than a known divergent series, it diverges.
• Ratio Test: Looks at the ratio of successive terms. If this ratio is less than one as terms advance, the series converges.
• Limit Comparison Test: Similar to the comparison test, but uses the limit of the ratio of the terms of two series for a conclusive result.

Even though a series like \( \Sigma a_n \) is convergent, applying tests is crucial when determining whether a related series like \( \sum \sin(a_n) \) converges.

Sine Function Properties

The sine function, \( \sin(x) \), is one of the fundamental trigonometric functions. Here are some properties, especially relevant in the context of convergence.

• Behavior Near Zero: For small values of \( x \), \( \sin(x) \approx x \). This is due to the sine function's linear behavior near zero, useful in approximations.
• Boundedness: The sine function varies between \(-1\) and \(1\) for any real number \( x \). This bounded nature implies that any manipulation or transformation of series using \( \sin \) will not exaggerate beyond these bounds.
• Periodic Nature: Sine is periodic with a period of \( 2\pi \). However, this property is more related to oscillatory behaviors than it is to convergence.

The property \( \sin(a_n) \approx a_n \) for small \( a_n \) gives an initial impression that similar convergence properties should hold. But subtle differences can arise when transitioning from \( a_n \) to \( \sin(a_n) \).

Counterexamples in Series Convergence

Counterexamples serve as powerful tools for understanding the limitations of assumptions in series convergence. They show us surprising outcomes where intuitive assumptions fail.

• Consider the series \( \sum a_n \), which is convergent. It may seem intuitive that transforming the series with a function like \( \sin \) should preserve convergent behavior, especially if \( \sin(a_n) \approx a_n \).
• Take the case where \( a_n = \frac{1}{n} \). The series \( \sum \frac{1}{n} \) is known as the harmonic series and it diverges.
• Even though each \( \sin(\frac{1}{n}) \approx \frac{1}{n} \) as \( n \) grows large, the transition from \( a_n \) to \( \sin(a_n) \) doesn’t solve the divergence issue present in the harmonic sequence.

Thus, counterexamples like these highlight that just because a series shares some properties with another, they do not necessarily share convergence. This emphasizes the importance of rigorous testing and deeper analysis.