Problem 10 - Single Variable Calculus

Question

Determine whether the \(p\) -series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2 / 3}} $$

Step-by-Step Solution

Answer

The given p-series is \(\sum_{n=1}^{\infty} \frac{1}{n^{2 / 3}}\), where \(p = 2/3\). Since \(p \le 1\), the p-series convergence test states that the series is divergent. Therefore, the given p-series is divergent.

Step 1: 1. Identify the value of \(p\)

In our given p-series, we have: $$ \sum_{n=1}^{\infty} \frac{1}{n^{2/3}} $$ So, the value of \(p\) in this case is \(2/3\).

Step 2: 2. Apply p-series convergence test

The p-series convergence test states that a p-series converges if \(p > 1\) and diverges if \(p \le 1\). In our case, we have \(p = 2/3\), which is less than or equal to 1.

Step 3: 3. Determine the series' convergence or divergence

Since our value of \(p\) is less than or equal to 1, according to the p-series convergence test, the given series diverges. Therefore, the given p-series \(\sum_{n=1}^{\infty} \frac{1}{n^{2 / 3}}\) is divergent.

Key Concepts

Series Divergence

A series is said to diverge when its terms do not approach a finite limit as they are summed to infinity. In simpler terms, the sum keeps increasing indefinitely and doesn't settle at a specific value. When studying series, it's important to understand whether the sum keeps growing. This is usually tested using various convergence tests, one of which is the p-series test. If a series diverges, knowing that it keeps expanding can help in understanding its behavior and potential applications, or deciding other mathematical methods to apply. In the context of the given exercise, recognizing that the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{2/3}} \) diverges because of its parameter means acknowledging that the series' sum grows without bound.

Convergence Test

Convergence tests are mathematical methods used to determine whether a series converges or diverges. These tests play a vital role in analyzing infinite series. Various tests include the p-series test, ratio test, and integral test, among others.For a series to converge, it must meet specific criteria dictated by these tests. The p-series test, for instance, determines convergence based largely on the value of \(p\) in the series. When performing these tests, especially for homework problems or in classroom situations, understanding their specifics can guide the problem-solving process.The convergence test used in our particular exercise is the p-series test, where we identify the value of \( p \), and then check whether it satisfies convergence conditions. This step helps us conclude directly whether our series diverges or converges.

P-Series

A p-series takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive real number. The behavior of a p-series is decisively influenced by the value of \( p \).

• If \( p > 1 \): The series converges.
• If \( p \leq 1 \): The series diverges.

Understanding p-series is essential because of how it simplifies analyzing certain infinite series. Whenever you encounter an infinite series whose terms can be expressed in the form \( \frac{1}{n^p} \), identifying the value of \( p \) allows the use of the p-series test to determine its behavior quickly.In the exercise, since \( p \) is \( \frac{2}{3} \), which is less than one, the series diverges by the p-series test.

Infinite Series

An infinite series is a sum of infinite terms, expressed usually as \( \sum_{n=1}^{\infty} a_n \), where each term is defined by a sequence \( a_n \). The fascinating but challenging aspect of an infinite series is determining what happens when you add infinitely many numbers. The sum of an infinite series can either approach a finite limit or continue to increase. This infinite behavior is why convergence tests are crucial in calculus and analysis. Some series, like geometric and telescoping series, have special conditions under which they converge.In our context, it's imperative to understand the infinite nature and apply known tests, such as the p-series convergence test, to draw conclusions about series behavior. With the given exercise asking to find convergence or divergence, acknowledging the series as infinite helps underscore why such analysis is useful.