Problem 64 - Thomas Calculus

Question

If \(\sum a_{n}\) is a convergent series of positive terms, prove that \(\sum \sin \left(a_{n}\right)\) converges.

Step-by-Step Solution

Answer

The series \(\sum \sin(a_{n})\) converges by the comparison test.

Step 1: Understand the Given Problem

We are given a series \(\sum a_{n}\) that converges and each \(a_{n}\) is positive. We are required to show that the series \(\sum \sin(a_{n})\) also converges.

Step 2: Knowledge of Sin Function

Recall that the sine function, \(\sin(x)\), for small positive values of \(x\) approximately equals \(x\). More precisely, \(\sin(x) \leq x\) for all \(x > 0\). This is because the function \(\sin(x)/x\) approaches 1 from below as \(x\) approaches zero.

Step 3: Compare Terms of Series

Since \(\sin(a_n) \leq a_n\) for each term in the series and each \(a_n > 0\), it follows that the new series \(\sum \sin(a_n)\) is term-wise smaller than the original series \(\sum a_n\). This is the critical comparison that lands the convergence argument.

Step 4: Applying the Comparison Test

The comparison test states that if \(\sum b_n\) converges and \(0 \leq a_n \leq b_n\) for all \(n\), then \(\sum a_n\) also converges. Since \(\sum a_n\) is convergent and \(\sin(a_n) \leq a_n\), the series \(\sum \sin(a_n)\) also converges by the comparison test.

Key Concepts

Sine Function

The sine function is a well-known trigonometric function, often denoted as \( \sin(x) \). It's important in mathematics because it describes the ratio of the length of the side opposite to an angle to the hypotenuse in a right-angled triangle. However, its significance becomes evident in calculus and series analysis due to some of its properties.

For small values of \( x \), the sine of \( x \) behaves very similarly to \( x \) itself. Specifically, we can approximate this peculiar behavior as \( \sin(x) \approx x \) when \( x \) is near zero. The mathematical property \( \sin(x) \leq x \) for all \( x > 0 \) is extremely useful when dealing with series. It shows that for small angles, the sine function's output is always equal to or less than the angle itself. This characteristic is key to establishing series convergence since it allows us to compare and bound terms very effectively by the sine function's direct input.

Comparison Test

The comparison test is a pivotal tool in determining the convergence of series. When dealing with series that are confusing or cumbersome, the comparison test simplifies the process by allowing a comparison of the unknown 'difficult' series to a 'simple' series whose convergence or divergence is already known.

The essence of the test lies in two conditions:

• If you have two series, \( \sum a_n \) and \( \sum b_n \), where every term of one series is less than or equal to a corresponding term of the other, i.e., \( 0 \leq a_n \leq b_n \), and
• If \( \sum b_n \) converges,

then by the comparison test, \( \sum a_n \) also converges. Conversely, if \( \sum a_n \) diverges, then \( \sum b_n \) must also diverge.

This test provides an assurance that if one series is "dominated" by a converging series, it must also converge; this property is what we used to show that \( \sum \sin(a_n) \) converges because it's term-for-term less than \( \sum a_n \), which already converges.

Convergent Series

A series is deemed convergent if the sum of its infinite sequence of terms approaches a single finite value. This concept is fundamental in calculus, especially in determining limits and understanding functions in deeper contexts.

For a given series \( \sum a_n \) to be convergent:

• The sequence of partial sums \( S_N = a_1 + a_2 + ... + a_N \) must get arbitrarily close to a specific number as \( N \) approaches infinity.
• This implies there is no room for the terms to increase over time; essentially, they should become smaller and add less and less to the total sum as you go further along.

Understanding convergence also points us towards necessary tests and conditions like the comparison test, ratio test, and others. These methods and conditions allow mathematicians to start validating whether more complex series will sum to a finite value or not, much like the sine series \( \sum \sin(a_n) \) in the exercise where its convergence was ratified by comparing it to another already known convergent series.