Problem 7 - Active Calculus

Question

In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the alternating series $$ 1-1+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{9}+\frac{1}{4}-\frac{1}{16}+\cdots $$ where the terms are selected alternately from the sequences \(\left\\{\frac{1}{n}\right\\}\) and \(\left\\{-\frac{1}{n^{2}}\right\\}\). a. Explain why the \(n\) th term of the given series converges to 0 as \(n\) goes to infinity. b. Rewrite the given series by grouping terms in the following manner: $$ (1-1)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{9}\right)+\left(\frac{1}{4}-\frac{1}{16}\right)+\cdots $$ Use this regrouping to determine if the series converges or diverges. c. Explain why the condition that the sequence \(\left\\{a_{n}\right\\}\) decreases to a limit of 0 is included in the Alternating Series Test.

Step-by-Step Solution

Answer

The nth term converges to 0; regrouped series suggests potential divergence; the decrease condition ensures proper term behavior leading to convergence.

Step 1: Show that the nth term converges to 0

Consider the sequences \(\frac{1}{n}\) and \(-\frac{1}{n^{2}}\). As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0 and \(-\frac{1}{n^{2}}\) also approaches 0. Therefore, the general nth term of the series, which is an alternating term from these two sequences, will also approach 0 as \(n\) goes to infinity.

Step 2: Group the terms of the series

Rewrite the series by grouping: \((1-1) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{9}\right) + \left(\frac{1}{4} - \frac{1}{16}\right) + \cdots\).

Step 3: Simplify each grouped term

Simplify each group in the series: \((1-1) = 0\) \(\left(\frac{1}{2} - \frac{1}{4}\right) = \frac{1}{4}\) \(\left(\frac{1}{3} - \frac{1}{9}\right) = \frac{2}{9}\) \(\left(\frac{1}{4} - \frac{1}{16}\right) = \frac{3}{16}\).

Step 4: Determine convergence or divergence

Consider the simplified terms: \(0, \frac{1}{4}, \frac{2}{9}, \frac{3}{16}, \cdots\). This sequence of positive terms tends to become smaller but does not necessarily sum to a finite value. As the denominator grows, the positive terms tend to be small but non-zero, implying potential divergence since the series forms a harmonic-like series.

Step 5: Explain the condition of the Alternating Series Test

The Alternating Series Test requires that the terms of the series decrease to 0. The provided series has parts that satisfy this as \(-\frac{1}{n^{2}}\) decreases, but \(\frac{1}{n}\) does not. This is why the test ensures that terms approaching zero indicate a potential convergence based on the smaller negative contributions dominating.

Key Concepts

Series Convergence

When we talk about **Series Convergence**, we're looking at whether adding up all the terms in a series will give us a finite number. This is important in mathematics because it helps us understand long-term behavior of sequences and series, rather than just individual terms or small subsets.

For a series \(\textstyle \sum a_n \) to converge, the sum of its terms as it progresses towards infinity must approach a specific number. This doesn't mean the individual terms approach a set value, but rather their cumulative addition trends towards convergence.

Take the given series, for example. To determine if it converges, we regroup the terms and simplify them: \(1 - 1 = 0 \), \( \frac{1}{2} - \frac{1}{4} = \frac{1}{4} \), \( \frac{1}{3} - \frac{1}{9} = \frac{2}{9} \). Each term is positive and decreasing but continues to grow smaller without reaching zero. This tells us the series might not sum to a finite number, hence hints at **divergence**.

Mathematical Sequences

To understand **Mathematical Sequences**, begin with the idea of *ordered lists* of numbers that follow a specific pattern. For instance, the sequences given in the exercise \( \frac{1}{n} \) and \( -\frac{1}{n^2} \) both consist of terms that decrease as \( n \) increases.

Here's a closer look:

• **Sequence \( \frac{1}{n} \)**: This is a *harmonic sequence*, where each term is the reciprocal of an integer. As \( n \) increases, \( \frac{1}{n} \) gets smaller and approaches **0**.
• **Sequence \( -\frac{1}{n^2} \)**: These are negative fractions where the numerator is always -1 and the denominator is the square of an integer. As \( n \) increases, \( -\frac{1}{n^2} \) also tends towards **0**.

Loving sequences means appreciating their predictable behaviors as they progress. Their convergence behavior often gives insight into the larger series they contribute to.

Harmonic Series Analysis

The **Harmonic Series** is a specific type of series known for its interesting properties related to convergence. Mathematically, it's expressed as:
\(\textstyle \sum \frac{1}{n} \) where \( n = 1, 2, 3, ...\).

Here's the catch: even though each term \( \frac{1}{n} \) gets smaller and smaller, the **total sum** never settles on a finite number. This is a classic case of **divergence**.

In the provided exercise, each component sequence - harmonic \( \frac{1}{n} \) and the quadratic \( \frac{1}{n^2} \) contribute to an overall behavior. Harmonic terms \( \frac{1}{n} \) decrease slower compared to negative quadratic terms \( -\frac{1}{n^2} \). Therefore, when we combine them in an alternating way, we have to be cautious. The positive harmonic terms might dominate leading to divergence unless controlled by stricter conditions.

Understanding the harmonic series is crucial because it shows how sequences can behave differently. Even as individual terms appear negligible, their accumulative sum can tell an opposite story. This insight is key to mastering concepts of series and sequence convergence.