Question

State whether the given series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n^{5}}$$

Short answer

The series converges because \( p = 5 > 1 \).

Step-by-step solution

Identify the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{1}{n^5} \). This is a p-series, which has the general form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In our case, \( p = 5 \).

Determine the Type of p-Series

For a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), the series converges if \( p > 1 \) and diverges if \( p \leq 1 \). Here, we have \( p = 5 \), which satisfies \( p > 1 \).

Conclude the Convergence

Since \( p = 5 \) and it is greater than 1, the given series \( \sum_{n=1}^{\infty} \frac{1}{n^5} \) is a convergent series by the properties of p-series.

Key concepts

Series Convergence

Series convergence refers to a property of infinite series where the sum of its terms approaches a specific value as the number of terms increases indefinitely. To understand convergence, one must consider the nature of the series itself.

A series is said to converge if the sequence of its partial sums converges to a limit. This means that as you add more terms, the total value approaches a particular number and doesn’t increase or decrease without bound. In simpler terms, a convergent series has a finite sum.

To determine if a series converges, mathematicians use various tests, such as the p-series test, which is applicable here. Understanding series convergence is critical, as it helps in analyzing how functions and sequences behave in more advanced mathematical topics. This concept is foundational for calculus, especially when dealing with integrals and infinite sums.

Infinite Series

An infinite series is the sum of an infinite sequence, or rather, the terms in the sequence keep going without end. Infinite series are a central part of calculus and mathematical analysis.

One key aspect of infinite series is that they can either converge or diverge, meaning their sum can approach a specific limit or fail to approach any limit. For example, the series \[ \sum_{n=1}^{\infty} \frac{1}{n^5} \]represents an infinite series where each term is the reciprocal of higher powers of integers.

When dealing with infinite series, the focus is not on the infinite number of terms, but rather on whether these terms add up to a finite number, which directly ties into the concept of convergence. Infinite series are crucial in representing functions through power series, and in defining systems of equations in linear algebra, among other applications.

Convergent Series

A convergent series is a type of infinite series where the sum of all terms results in a finite number. For a series to be convergent, its partial sums should get closer and closer to a particular number as more terms are added.

A sign that a series might converge is if its terms become very small as the series progresses. This is because smaller terms contribute less to the overall sum, helping it settle towards a particular value. To confirm convergence, specific tests and criteria must be met, such as the p-series test.

Take the series \[ \sum_{n=1}^{\infty} \frac{1}{n^5} \]for example. Here, the value of p is greater than 1, which according to the properties of a p-series indicates that the series converges. Recognizing convergent series assists in understanding many mathematical functions and phenomena, as it ensures the series can approximate values with great accuracy.