Problem 12 - Single Variable Calculus

Question

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

Step-by-Step Solution

Answer

The given p-series is \(\sum_{n=1}^{\infty} \frac{1}{n^e}\), where \(p=e\). Since \(e \approx 2.718 > 1\), according to the p-series convergence theorem, the p-series converges.

Step 1: Identify the value of p in the p-series

The given p-series is \(\sum_{n=1}^{\infty} \frac{1}{n^{e}}\). Here, \(e\) is the power to which \(n\) is raised, so in this case, \(p = e\).

Step 2: Apply the p-series convergence theorem

According to the p-series convergence theorem, a p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges when \(p > 1\) and diverges when \(p \le 1\). Since in this case, \(p = e\), we will compare \(e\) to 1. We know that the value of \(e\) (Euler's number) is approximately 2.718. This can be written as \(e \approx 2.718\), and therefore, \(e\) is greater than 1.

Step 3: Determine if the series converges or diverges

Since \(e > 1\), based on the p-series convergence theorem, the given p-series \(\sum_{n=1}^{\infty} \frac{1}{n^{e}}\) converges. Therefore, the p-series converges.

Key Concepts

series convergence

In mathematics, determining whether a series converges or diverges is an important concept. Convergence refers to the behavior of a series such that the sum of its terms approaches a particular value, while divergence means the sum grows indefinitely. For many classes of series, particularly p-series, specific criteria help us understand convergence.
For a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), the series converges if the exponent \( p > 1 \). It diverges if \( p \le 1 \). This simple rule simplifies the process of checking convergence for these types of series.
It's worth noting that series convergence is not solely about p-series. Various tests exist, like the comparison test, ratio test, and integral test, to help analyze different series types. Grasping the idea of convergence is essential for deeper mathematical studies and real-world applications.

Euler's number

Euler's number, denoted as \( e \), is a fundamental constant in mathematics, approximately valued at 2.71828. It is important in various mathematical contexts, including series and calculus.
Euler's number has unique properties. For instance, the function \( e^x \) (the exponential function) has a derivative equal to the function itself, making it crucial for solving differential equations.

• Approximately equal to 2.71828
• Key role in exponential growth models
• Solution to many natural logarithm equations

In the context of the p-series problem, \( e \) determines the power to which \( n \) is raised. Since \( e \) is greater than 1, it directly contributes to the series' convergence under the p-series theorem. Understanding Euler's number enhances comprehension of exponential growth, compound interest, and more.

convergence theorem

The convergence theorem provides a guideline to decide whether a p-series converges based on its exponent \( p \). This theorem states that a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) will converge if \( p > 1 \) and diverge if \( p \le 1 \).
The application of this theorem is simple yet powerful, offering a direct way to analyze series with forms \( \frac{1}{n^p} \), especially when dealing with infinite limits. In our example, knowing that \( e \approx 2.718 \) allows us to apply the theorem easily for a quick determination of convergence.
This theorem not only aids in theoretical calculations but also applies to practical scenarios where p-series might model natural phenomena or financial models. Its straightforward application makes it a valuable tool for students and mathematicians alike, allowing for efficient analysis of series and patterns.