Problem 47 - Single Variable Calculus

Question

In Exercises \(47-50\), determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) converge absolutely, then \(\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)\) converges absolutely.

Step-by-Step Solution

Answer

The statement "If \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) converge absolutely, then \(\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)\) converges absolutely" is true. This can be proven using the triangle inequality \(|a_{n}+b_{n}|\leq |a_{n}|+|b_{n}|\), and the Comparison Test, which shows that since \(\sum_{n=1}^{\infty}(|a_{n}|+|b_{n}|)\) converges, so must \(\sum_{n=1}^{\infty}|a_{n} + b_{n}|\), proving that the sum of the series \(\sum_{n=1}^{\infty}(a_{n}+b_{n})\) converges absolutely.

Step 1: Recall the definition of absolute convergence

A series \(\sum_{n=1}^{\infty} c_{n}\) is said to converge absolutely if the series of the absolute values of its terms, \(\sum_{n=1}^{\infty} |c_{n}|\), converges. Given: \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) converge absolutely.

Step 2: Sum of the series

Our goal is to determine if the series \(\sum_{n=1}^{\infty}(a_{n}+b_{n})\) converges absolutely. To show this, we need to find if the series \(\sum_{n=1}^{\infty}|a_{n} + b_{n}|\) converges.

Step 3: Use the Triangle Inequality

We apply the triangle inequality to the series \(\sum_{n=1}^{\infty}|a_{n}+b_{n}|\). Recall that the triangle inequality states that \(|x+y| \leq |x| + |y|\) for any real numbers x and y. Hence, \(|a_{n} + b_{n}| \leq |a_{n}| + |b_{n}|\) Summing both sides from n=1 to infinity, we get: \(\sum_{n=1}^{\infty}|a_{n} + b_{n}| \leq \sum_{n=1}^{\infty}(|a_{n}| + |b_{n}|)\)

Step 4: Verify the convergence of the series

From the given information, we know that \(\sum_{n=1}^{\infty} |a_{n}|\) and \(\sum_{n=1}^{\infty} |b_{n}|\) are convergent. Since the sum of convergent series is also convergent, it follows that \(\sum_{n=1}^{\infty}(|a_{n}| + |b_{n}|)\) converges. By the Comparison Test, if a non-negative series converges and is greater than or equal to another non-negative series, the other series also converges. Since the series \(\sum_{n=1}^{\infty}|a_{n} + b_{n}|\) is non-negative and less than or equal to the convergent series \(\sum_{n=1}^{\infty}(|a_{n}| + |b_{n}|)\), it also converges. Hence, the series \(\sum_{n=1}^{\infty}(a_{n} + b_{n})\) converges absolutely. The statement is true.

Key Concepts

series

A series is essentially the sum of terms in a sequence, which can go on indefinitely. In mathematics, we often see series expressed using summation notation, such as \( \sum_{n=1}^{\infty} a_n \), which reads as "the sum of \( a_n \) from \( n = 1 \) to infinity." This notation captures the idea that we're summing an infinite number of elements.
Understanding whether a series converges or diverges is crucial. A convergent series approaches a specific number as more and more terms are added. On the other hand, a divergent series does not settle to a fixed value.
When working with series, keep in mind:

• Each term in the series is part of a sequence.
• We're interested in the behavior of the entire sum as it continues to infinity.

Finding out if a series converges or diverges helps in understanding its overall behavior.

triangle inequality

The triangle inequality is a fundamental concept in mathematics, especially useful in analysis and geometry. It states that for any real numbers \( x \) and \( y \), the absolute value of their sum is less than or equal to the sum of their absolute values:
\[ |x + y| \leq |x| + |y| \] This inequality is similar to the idea that the shortest distance between two points is a straight line, symbolized geometrically by a triangle's sides.
In terms of series, applying the triangle inequality helps in comparing sums of absolute values. We use it to establish bounds, particularly when we want to prove absolute convergence.
By utilizing this inequality, if you have a sum of two series, such as \( \sum_{n=1}^{\infty} (a_n + b_n) \), the sum's absolute value will not exceed the sum of the individual absolute series. This is a handy tool when demonstrating convergence properties of series.

comparison test

The comparison test is a method in calculus to determine whether a series converges. It involves comparing the series in question to another series whose convergence behavior is known.
The basic idea is quite straightforward:

• Suppose you have two series, \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \), where all terms are non-negative.
• If \( a_n \leq b_n \) for all \( n \) and the series \( \sum_{n=1}^{\infty} b_n \) is convergent, then \( \sum_{n=1}^{\infty} a_n \) is also convergent.

This test helps to establish convergence by showing that a series is dominated by a known convergent series. Conversely, if \( b_n \) diverges and \( a_n \geq b_n \), then \( a_n \) also diverges.
Applying the comparison test is an effective way to validate the convergence of complex series, making it a crucial tool in mathematical analysis.

convergent series

In mathematics, a convergent series is one where the sum of its infinite terms approaches a specific limit. Convergence means that as you add more and more terms, the total gets closer to a particular number, rather than veering off to infinity.
Here are some key points about convergent series:

• A series \( \sum_{n=1}^{\infty} a_n \) converges if there's a limit \( L \) such that the partial sums approach \( L \) as \( n \) grows.
• Absolute convergence strengthens this concept. If the series of the absolute values, \( \sum_{n=1}^{\infty} |a_n| \), also converges, the original series is absolutely convergent.

Absolute convergence is a powerful idea because it guarantees the series will behave nicely. It's a stronger form of convergence and implies traditional convergence.
Understanding convergence is essential for solving problems involving infinite processes and ensures that mathematical statements about sums hold true.