Question

Does the series \(\sum_{n=1}^{\infty} \frac{1}{n^{5 / 4}}\) converge or diverge?

Short answer

The series converges.

Step-by-step solution

Series Identification

Recognize that the given series is \( \sum_{n=1}^{\infty} \frac{1}{n^{5 / 4}} \). This is a p-series, where the general form is \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Identifying the type of series is crucial for determining convergence.

Determine p-value

Identify the value of \( p \) in the series \( \sum_{n=1}^{\infty} \frac{1}{n^{5 / 4}} \). Here, \( p = \frac{5}{4} \). This is essential for applying the p-series test.

Apply p-series Test

According to the p-series test, a series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \). Here, \( \frac{5}{4} = 1.25 \), which is greater than 1.

Conclusion on Convergence

Based on the p-series test, since \( \frac{5}{4} > 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{n^{5 / 4}} \) converges. This is the final conclusion drawn from evaluating the series.

Key concepts

p-series

A p-series is a type of infinite series that comes in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In this notation, \( p \) is a positive constant that significantly influences the behavior of the series.
Understanding the structure of a p-series helps in determining whether it converges or diverges.
For example, common instances of p-series include the harmonic series, \( \sum_{n=1}^{\infty} \frac{1}{n} \), which corresponds to \( p = 1 \), and famously diverges. Another example is the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), known as the Basel problem, which converges because \( p > 1 \).
To apply tools for analyzing series, it's crucial to first identify the series as a p-series by recognizing its pattern. This recognition provides a pathway to use specific tests like the p-series test effectively.

p-series test

The p-series test is a straightforward criterion used to determine the convergence of p-series. It states:

• A p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \).
• Conversely, it diverges if \( p \leq 1 \).

Understanding this test makes it easy for students to assess a wide variety of series just by identifying the value of \( p \).
For instance, if you have a series with \( p = \frac{5}{4} \), since \( 1.25 > 1 \), the series converges.
The p-series test is simple yet powerful, acting as a gatekeeper in series analysis by helping to quickly determine the convergence criteria without exhaustive calculations. Employing this test is about recognizing the power of the denominator and checking it against the critical boundary of 1.

convergence

Convergence in the context of series refers to the behavior of the terms of the series as you sum an infinite number of them. A series converges if the sum approaches a finite limit.
When a series does not converge, it is said to diverge, meaning the total sum increases indefinitely or does not settle into a specific value.
Several tests and techniques exist to determine convergence, with the p-series test being one of the most straightforward for series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \).
Determining convergence often involves identifying patterns or structures within the series, and then applying the tests that take into account the series' specific properties. Knowing whether a series converges is crucial in fields like numerical analysis and mathematical modeling, where the sum of infinite terms is used to represent real-world phenomena or solve problems.