Problem 33 - Calculus: Early Transcendentals

Question

For what value of \( p \) is each series convergent? $$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n}{n + p} $$

Step-by-Step Solution

Answer

The series converges for \( p > 0 \).

Step 1: Understanding the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n + p} \). This is an alternating series due to the term \((-1)^n\), which changes sign each time \(n\) increases by 1. To determine convergence, we need to check the conditions for alternating series.

Step 2: Applying the Alternating Series Test

The alternating series test states that the series \( \sum (-1)^n a_n \) converges if the sequence \( a_n \) (the terms without the alternating sign) decreases monotonically and approaches 0 as \( n \to \infty \). Here, \( a_n = \frac{1}{n + p} \). We will check both conditions for \( a_n \).

Step 3: Ensuring the Sequence Decreases

For \( \frac{1}{n + p} \) to be decreasing as \( n \) increases, \( n + p \) must increase. Since \( n \) increases naturally, we require \( p > 0 \) so that \( n + p \) remains positive and increasing, ensuring \( \frac{1}{n + p} \) decreases.

Step 4: Checking Limit Approaches Zero

Calculate \( \lim_{n \to \infty} \frac{1}{n + p} \). For any \( p \geq 0 \), \( \lim_{n \to \infty} \frac{1}{n + p} = 0 \), satisfying the second condition of the alternating series test.

Step 5: Conclusion on Convergence

The series converges if \( a_n = \frac{1}{n + p} \) is both decreasing and approaches zero. We found \( p > 0 \) ensures these conditions. Therefore, the series is convergent when \( p > 0 \).

Key Concepts

Alternating Series Test

The alternating series test is a crucial tool for determining whether certain series converge. This concept applies to series where the terms alternate between positive and negative. In simpler terms, an alternating series is one that changes its sign with each term. A typical form for such a series is

• \( \sum (-1)^n a_n \)

To conclude convergence using the alternating series test, two conditions must be met:

• The terms \( a_n \) must decrease as \( n \) increases, meaning the series is made up of ever-smaller positive numbers.
• The sequence \( a_n \) approaches zero as \( n \) approaches infinity. Simply put, the terms become vanishingly small.

For the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n + p} \), these conditions help determine the value of \( p \) for which the series converges. This simple test takes a seemingly complex series and breaks down its convergence into manageable conditions.

Convergence

Convergence refers to the behavior of a series as more and more terms are added. The idea is to see whether the series approaches some definite value or not. If it does, the series is said to converge. In the context of alternating series, like \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n + p} \), understanding convergence means verifying whether the terms' influences balance out eventually, settling around a specific value despite the constant sign changes.The convergence is investigated using various tests, one of which is the alternating series test. With this test, determining convergence involves checking monotonic decrease and zero limits of the sequence \( a_n \). It's crucial to recognize that while some series converges, others may diverge, meaning they do not settle towards a particular limit.

Series Convergence

Series convergence is the broader process of evaluating whether a series converges. Each type of series may require a different approach or test for determining convergence. For example, alternating series often rely on specific tests designed for their unique properties. Some general strategies for determining convergence include:

• Using the alternating series test for series with alternating terms.
• Checking for monotonic decrease and approaching zero in terms.
• Applying comparison tests or integral tests if applicable.

For our series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n + p} \), the alternating series test is suitable. Once each requisite condition is verified, particularly how \( p \)'s value affects the series, conclusions about convergence can be confidently drawn.

Calculus

Calculus plays a pivotal role in understanding advanced mathematical concepts like series convergence. It provides the tools necessary to evaluate scenarios where non-intuitive patterns arise, such as alternating series. For instance:

• Using limits to inspect how individual terms of a series behave as \( n \to \infty \).
• Formulating conditions under which a series converges using analytical methods.
• Applying derivative and integration techniques to explore changes tightly woven into the behavior of the series.

The role of calculus becomes evident in finding when an alternating series like \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n + p} \) converges by analyzing its terms and employing tests and limits directly stemming from fundamental calculus principles. This blend of theory and calculation provided by calculus offers a structured pathway to tackling diverse series questions.