Question

Determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} \frac{1}{n^{n}}$$

Short answer

The series converges by the Root Test.

Step-by-step solution

Identify the General Term

The general term of the series is given by \( a_n = \frac{1}{n^n} \). We will apply the Root Test to this series.

Apply the Root Test

According to the Root Test, we examine the expression \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). For our series, this becomes \( \lim_{n \to \infty} \sqrt[n]{\frac{1}{n^n}} = \lim_{n \to \infty} \left(\frac{1}{n^n}\right)^{1/n} = \lim_{n \to \infty} \frac{1}{n} \).

Evaluate the Limit

Compute the limit: \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

Interpret the Root Test Result

Since the limit \( L = 0 \) and \( L < 1 \), the Root Test concludes that the series \( \sum_{n=1}^{\infty} \frac{1}{n^n} \) converges.

Key concepts

Convergence of series

The concept of convergence in series is crucial in mathematics because it determines whether a series adds up to a finite number. A series converges if the sum of its terms approaches a specific finite value as more and more terms are added. Understanding convergence helps us evaluate whether a seemingly endless sum can be represented simply, like adding infinitely many fractions that still sum to a number.

• A series can be divergent, meaning it grows without bound or doesn't settle to a number.
• Convergence implies stability in the sum.
• Different tests can be used to check convergence, such as the Root Test, Ratio Test, and Comparison Test.

For example, the Root Test helps by examining the behavior of the series' terms as the series progresses toward infinity.
In our case, understanding the convergence of the series \( \sum_{n=1}^{\infty} \frac{1}{n^n} \) tells us it doesn't blow up as we add more terms.

General term of a series

The general term of a series is a formal representation of each term in the series, typically in terms of \( n \), which denotes the term number. It gives us a blueprint of the sequence that comprises the series.

• The general term helps in applying tests to determine convergence or divergence.
• Understanding the general term is pivotal to analyzing the series for mathematical operations.

For instance, in the series \( \sum_{n=1}^{\infty} \frac{1}{n^{n}} \), the general term is \( a_n = \frac{1}{n^n} \).
Each term becomes progressively smaller because \( n^n \) grows much faster than any constant or linear function involving \( n \).
This insight into the structure of the general term can immediately suggest convergence as we move into applying various tests.

Limit evaluation

Evaluating limits is an integral part of calculus and is particularly useful in series analysis. It allows us to explore the behavior of a function as the input approaches a specific value, often infinity in series.

• Limits help determine how terms of the series behave as they stretch towards infinity.
• They are essential in applying tests like the Root Test, where the limit of the general term guides the convergence decision.

When performing limit evaluation using the Root Test, we compute \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \), which simplifies how a series term 'contracts' in size.
For example, with \( \sum_{n=1}^{\infty} \frac{1}{n^n} \), evaluating \( \lim_{n \to \infty} \frac{1}{n} = 0 \) tells us the terms shrink quickly enough for the sum to converge.
This limit's approach allows us to confidently conclude the series' behavior without direct summation.