Problem 1 - Thomas Calculus

Question

In Exercises \(1 - 14 ,\) determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { \sqrt { n } } $$

Step-by-Step Solution

Answer

The series converges by the Alternating Series Test.

Step 1: Identify the Alternating Series

The given series is \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{\sqrt{n}} \). This is an alternating series because of the \((-1)^{n+1}\) factor that alternates the sign of each term.

Step 2: Apply the Alternating Series Test

The Alternating Series Test requires two conditions: 1) the absolute value of terms, \(b_n = \frac{1}{\sqrt{n}}\), should be decreasing, and 2) \(b_n \to 0\) as \(n \to \infty\).

Step 3: Check if Sequence is Decreasing

To check if \(b_n = \frac{1}{\sqrt{n}}\) is decreasing, note that for \(n < m\), \( \sqrt{n} > \sqrt{m} \) implies \( \frac{1}{\sqrt{n}} < \frac{1}{\sqrt{m}} \). Thus, \(b_n\) is decreasing.

Step 4: Check the Limit of b_n

Calculate the limit \(\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0\). Since this limit is 0, the second condition of the Alternating Series Test is satisfied.

Step 5: Conclude Convergence

Since both conditions of the Alternating Series Test are satisfied (the terms decrease and approach zero), the series converges by the Alternating Series Test.

Key Concepts

Alternating Series Test

The Alternating Series Test is a simple yet powerful tool to determine if a series with alternating signs converges. When you see a series with a \((-1)^{n+1}\) or similar notation, it's likely to be alternating. This means the series has terms with signs that switch from positive to negative as you progress through the sequence.
This test requires checking two main conditions to confirm convergence:

• The terms of the sequence, without considering their signs, must consistently decrease in absolute value.
• The terms must approach zero as the index goes to infinity.

Once these conditions are verified, the series is convergent. In our example, the series is \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{\sqrt{n}}\), which is alternating because of the \((-1)^{n+1}\) factor. The next step is to see if it converges.

Convergence

Convergence in the context of an alternating series implies that the series adds up to a specific finite value, even if the series continues indefinitely. When the series terms meet the conditions of the Alternating Series Test, you can conclude that it converges.
For the provided series, \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{\sqrt{n}}\), convergence was confirmed using the test. This means that if we keep adding more and more terms from this sequence, it will converge, or get closer and closer, to a fixed number. The testing process ensures that the series doesn't just bounce around or increase without bound, but settles toward a single value.

Alternating Series Conditions

The conditions necessary for applying the Alternating Series Test revolve around two critical points: decreasing sequence terms and a zero limit.

1. **Decreasing Sequence:** The sequence of the absolute values of the terms must be decreasing. The sequence \(b_n = \frac{1}{\sqrt{n}}\) was shown to be decreasing because as n gets larger, \(\sqrt{n}\) increases, making \( \frac{1}{\sqrt{n}}\) smaller as n increases.2. **Zero Limit:** The terms of the sequence must approach zero. This was confirmed when we calculated \(\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0\).
These conditions ensure the series fulfills the requirements of the Alternating Series Test and affirms convergence.

Sequence Analysis

Sequence analysis involves evaluating the behavior of the terms in a sequence, which is crucial for understanding series like our example. By analyzing the term structure, we can predict the convergence behavior.

For the sequence \(b_n = \frac{1}{\sqrt{n}}\):

• We observed that it is decreasing. This observation came from knowing that the square root of an increasing n naturally becomes larger, leading to a smaller fraction for higher n.
• The evaluation of the limit as n approaches infinity was key. As n grows, the fraction goes to zero, meaning b_n shrinks towards zero.

This step-by-step analysis clarifies why the Alternating Series Test conditions are satisfied for the given sequence.