Question

(a) Prove: If \(\sum a_{n}\) converges and \(\sum a_{n}^{2}=\infty,\) then \(\sum a_{n}\) converges conditionally. (b) Give an example of a series with the properties described in (a).

Short answer

Question: Prove that a series converges conditionally given that the series converges and its square converges to infinity. Then, provide an example of such a series. Answer: To prove conditional convergence, we used the Cauchy-Schwarz inequality and deduced that due to the given information, the series of absolute values must diverge. An example of a series that meets these conditions is the alternating harmonic series \((-1)^{n+1} \frac{1}{\sqrt{n}}\), as it converges by the Alternating Series Test, has a square that converges to infinity, and has a series of absolute values that diverge.

Step-by-step solution

(a) Proving Conditional Convergence

Since we are given that \(\sum a_{n}\) converges, we know that \(\lim_{n \to \infty} a_{n} = 0\). We want to prove that \(\sum |a_{n}|\) diverges. To do so, we utilize the provided information that \(\sum a_{n}^{2} = \infty\). Using the Cauchy-Schwarz inequality, we can compare the series \(\sum |a_{n}|\) and \(\sum a_{n}^2\). Consider the series \(\sum |a_{n}|\) and \(\sum 1\). Applying the Cauchy-Schwarz inequality to both series: $$(\sum_{n}|a_{n}|)^2 \leq \sum_{n}a_{n}^2 \cdot \sum_{n}1$$ Since we know that \(\sum a_{n}^2 = \infty\), we can deduce that \(\sum |a_{n}|\) must be divergent, as any non-infinity value squared would lead to a finite number. Therefore, \(\sum a_{n}\) converges conditionally since \(\sum a_{n}\) converges and \(\sum |a_{n}|\) diverges.

(b) Providing an Example

To find an example of a series meeting the requirements, consider the following series: $$a_{n} = (-1)^{n+1} \frac{1}{\sqrt{n}}$$ Let's check the given conditions: 1. \(\sum a_{n}\) converges This series is an alternating series, and we can check for its convergence using the Alternating Series Test. The sequence \(\frac{1}{\sqrt{n}}\) is monotonically decreasing and has a limit of 0 as \(n \rightarrow \infty\). Therefore, the series \(\sum a_{n}\) converges by the Alternating Series Test. 2. \(\sum a_{n}^{2} = \infty\) Let's find \(a_{n}^2\): $$a_{n}^2 = \left((-1)^{n+1} \frac{1}{\sqrt{n}}\right)^2 = \frac{1}{n}$$ Now, \(\sum a_{n}^2\) is a harmonic series with p=1, and we know that the p-series converges for \(p > 1\). In this case, \(p = 1\), so the series \(\sum a_{n}^2 = \infty\). 3. \(\sum |a_{n}|\) diverges Since \(\sum a_{n}^2 = \infty\), we have already shown that \(\sum |a_{n}|\) diverges when we proved conditional convergence in part (a). Thus, the series \(a_{n} = (-1)^{n+1} \frac{1}{\sqrt{n}}\) satisfies all the given conditions and serves as an example with the properties described in part (a).

Key concepts

Convergent Series

When we speak of a convergent series in mathematics, we refer to an infinite series \(\sum a_n\) that approaches a specific value as more and more terms are added. In other words, there exists a finite limit L such that as n, the number of terms, increases, the partial sums \(\sum_{i=1}^n a_i\) come arbitrarily close to L. Convergence is a fundamental concept in analysis because it can determine whether an infinite sum has a finite result.

One common test to determine the convergence of a series is the 'Alternating Series Test', used when the series alternates between positive and negative terms. Another method involves the 'p-series' test, which can be used for series of the form \(\sum \frac{1}{n^p}\), where p is a constant. The key criterion here is that if p is greater than 1, the p-series converges; otherwise, it diverges. Convergent series have numerous applications in various fields of science and engineering, especially in calculating sums that would otherwise be incalculable due to their infinite nature.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a powerful tool in mathematical analysis, particularly when dealing with series and integrals. It establishes that for any two sequences of real numbers, the square of the sum of the products of these sequences is less than or equal to the product of the sums of each sequence squared. Mathematically, for any two sequences \(a_n\) and \(b_n\), the inequality is written as:
\[\left(\sum_{n} a_n b_n\right)^2 \leq \left(\sum_{n} a_n^2\right)\left(\sum_{n} b_n^2\right)\]

It holds an essential place in the study of series, allowing us to draw conclusions about convergence and divergence of infinite sums. In the context of the provided exercise, the Cauchy-Schwarz inequality helps us establish the divergence of a series by comparing it to another series where the nature of convergence or divergence is already known. This conceptual bridge facilitates a deeper understanding of the series' behavior and aids in proving conditional convergence.

Divergent Series

In contrast to convergent series, a divergent series is one where the partial sums do not approach any finite limit as more terms are added. This means that either the series approaches infinity, oscillates between different values, or does not settle on a particular value at all. A simple example of a divergent series is the harmonic series \(\sum \frac{1}{n}\), which increases without bound as n grows larger.

Divergence can be demonstrated through various tests and comparisons, one of which is–as introduced in the Cauchy-Schwarz inequality section–comparing the series in question with another series known to be divergent. Understanding the divergence of series is crucial as it indicates that the infinite sum cannot represent a finite quantity and that using the series to calculate a physical quantity might not yield a meaningful result.

Alternating Series Test

The Alternating Series Test is specifically devised for series that alternate between positive and negative terms. It states that an alternating series \(\sum (-1)^{n} a_n\) converges if two conditions are met: firstly, the absolute values of the terms \(a_n\) must decrease monotonically, meaning each term is less than or equal to the previous term; secondly, the limit of \(a_n\) as n approaches infinity must be zero.

Applying this test to an alternating series gives a straightforward method for determining its convergence without necessarily having to consider other more complex convergence tests. In our exercise, the Alternating Series Test is used for a series that conforms to an alternating sign pattern, ensuring that, despite the series having infinite terms, it summates to a finite value.

p-Series

A p-series is a specific type of series with a general term of the form \(\frac{1}{n^p}\), where 'p' is a constant real number. The convergence or divergence of a p-series depends strictly on the value of p. As a rule of thumb, if p is greater than 1, the series converges; conversely, when p is less or equal to 1, the series diverges.

In our exercise example, we see a p-series with \(p = 1\), which is known to diverge. Recognizing p-series and understanding their convergence properties are incredibly useful across various contexts within mathematics, including integral tests or comparisons between series, and they serve as a standard example for categorizing other series' behaviors.