Question

(a) Verify Corollary 4.3.7 for the convergent series \(\sum 1 / n^{p}(p>1) .\) HINT: See the proof of Theorem 4.3.10. (b) Verify Corollary 4.3.7 for the convergent series \(\sum(-1)^{n} / n\).

Short answer

Question: Verify Corollary 4.3.7 for the convergent series (a) \(\sum 1 / n^{p}(p>1)\) and (b) \(\sum(-1)^{n} / n\).

Step-by-step solution

Recall the p-series test

The p-series test states that the series \(\sum \frac{1}{n^p}\) converges if \(p>1\) and diverges if \(p\leq 1\). In this exercise, we are given \(p>1\), so the series converges.

Refer to Theorem 4.3.10

Using the hint given in the exercise, we can look at the proof of Theorem 4.3.10 to see if we can apply any part of it to this series. Try to understand the proof details and apply the relevant steps for the given series.

Verify the corollary

After understanding the relevant steps from the proof of Theorem 4.3.10, apply those steps to the series \(\sum \frac{1}{n^p}\) and show that the corollary is verified for this series. Part (b): Verify Corollary 4.3.7 for the convergent series \(\sum(-1)^{n} / n\).

Recall the alternating series test

An alternating series is of the form \(\sum (-1)^n a_n\) where \(a_n \ge 0\). The alternating series test states that an alternating series converges if the terms \(a_n\) are decreasing and approaching to \(0\) as \(n \rightarrow \infty\). In this case, \(a_n = \frac{1}{n}\), which satisfies these conditions.

Refer to Theorem 4.3.10 again

As we did in part (a), refer to the proof of Theorem 4.3.10 and find relevant steps that can be applied to the alternating series \(\sum \frac{(-1)^n}{n}\).

Verify the corollary

After applying the relevant steps from the proof of Theorem 4.3.10 to the series \(\sum \frac{(-1)^n}{n}\), show that the corollary is verified for this convergent series as well.

Key concepts

p-series test

The p-series test is an essential tool for determining the convergence of series in the form of \( \sum \frac{1}{n^p} \). For convergence, it is important that the parameter \( p \) is greater than 1. This implies that the terms of the series decrease at a rate sufficient enough for the series' sum to have a finite value.
This can be intuitively understood as follows:

• When \( p > 1 \), the denominator \( n^p \) grows faster than the numerator, making the terms smaller as \( n \) increases.
• This results in the infinite sum having a limit as \( n \) approaches infinity.

If \( p \leq 1 \), the terms of the series do not diminish fast enough, leading the series to diverge, as they fail to approach zero quickly. Understanding this test is fundamental to verifying whether a given series converges or diverges, as seen in the exercise through Corollary 4.3.7.

alternating series test

The alternating series test evaluates series that alternate in sign, usually represented as \( \sum (-1)^n a_n \), where \( a_n \geq 0 \). For such a series to converge, two primary conditions must be satisfied:

• The sequence \( a_n \) must be monotonically decreasing, meaning each term is less than or equal to the previous term.
• The terms \( a_n \) must approach zero as \( n \) approaches infinity.

For example, in the exercise, the series \( \sum (-1)^n \frac{1}{n} \) meets these criteria:

• The terms \( \frac{1}{n} \) decrease as \( n \) increases. This is because as the denominator \( n \) becomes larger, the value of the fraction gets smaller.
• The terms approach zero, satisfying the second condition of the alternating series test.

Thus, using this test confirms that the given alternating series converges, consistent with Corollary 4.3.7.

Theorem 4.3.10

Theorem 4.3.10 plays a pivotal role in confirming the convergence of series through detailed verification steps. While the exercise does not provide the specific details of this theorem, its application involves examining the behavior of series through rigorous proof.
Theorem 4.3.10 might demonstrate how convergence can be established by comparing the series to known convergent series or by using mathematical properties such as limits and inequalities. In the context of this exercise:

• Parts of this theorem can be used to analyze the series \( \sum \frac{1}{n^p} \) to ensure it satisfies all conditions for convergence, as utilized in the proof.
• It also provides a framework for evaluating alternating series like \( \sum (-1)^n \frac{1}{n} \), by offering methodologies that underscore the convergence verified by the alternating series test.

The application of Theorem 4.3.10 ensures a robust understanding and the correct application of convergence tests, bridging intuitive ideas with formal mathematical proofs.