Question

State whether the given series converges or diverges. $$\sum_{n=1}^{\infty} n^{-4}$$

Short answer

The series converges because it is a p-series with \( p = 4 > 1 \).

Step-by-step solution

Identify the Type of Series

The series given is \( \sum_{n=1}^{\infty} n^{-4} \). This can be identified as a **p-series**, which is of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p = 4 \).

Determine Conditions for Convergence of a p-series

For a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), it is known that the series converges if \( p > 1 \). If \( p \leq 1 \), the series diverges.

Apply the Convergence Test

In this case, \( p = 4 \). Since \( 4 > 1 \), by the p-series test, \( \sum_{n=1}^{\infty} n^{-4} \) converges.

Key concepts

Series Convergence

When we study sequences and series in calculus, series convergence is a crucial concept. It helps us determine whether a series approaches a finite value or not. A series is essentially the sum of the terms of an infinite sequence. To determine if a series converges, we must check if the sum approaches a specific limit. If the sum does reach a limit as the number of terms increases indefinitely, then we say the series converges.
Conversely, if the sum doesn't approach any limit, the series diverges. This provides a foundation for understanding many areas in mathematics and science where infinite processes occur. Series convergence is important for understanding the behavior of functions and is used in various applications such as engineering, physics, and economics.

p-Series Test

The p-series test is one of the simplest tests to check for convergence or divergence of a series. It specifically applies to series of the form:\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]Here, the variable \( p \) must be a constant and serves as a critical component in determining the convergence of the series. This is because the value of \( p \) directly dictates the behavior of the series:

• If \( p > 1 \), the p-series converges. This means the sum of the series will approach a specific finite limit.
• If \( p \leq 1 \), the series diverges, meaning the sum will increase indefinitely without approaching any limit.

This test is particularly useful for quickly assessing basic series and is a great starting point for deeper analysis in series behavior.

Convergence Criteria

Understanding the criteria for convergence of a series is essential for applying the correct tests. For a p-series, the main criterion revolves around the value of \( p \) in the expression \( \frac{1}{n^p} \). Since our goal is to determine whether the series converges or diverges, knowing the target \( p \) value is crucial:1. **p > 1**: If the exponent \( p \) is greater than 1 in the function \( \frac{1}{n^p} \), the series converges. This is because as \( n \) increases, the terms become significantly smaller, leading the sum towards a finite value.2. **p \leq 1**: If \( p \) is 1 or less, the series diverges. Here, the terms decrease at a slower rate, preventing the sum from approaching a specific limit.This helps mathematicians and students determine the behavior of infinite series with ease. These simple rules make the p-series test highly efficient in solving numerous mathematical problems.