Problem 16 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^n} \)

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Identify the Series

We are given the series \( \sum_{n = 1}^{\infty} \frac{1}{n^n} \). This is an infinite series where each term is \( a_n = \frac{1}{n^n} \). Our goal is to determine if this series converges or diverges.

Step 2: Apply the Ratio Test

The Ratio Test is a common method to determine the convergence of a series. According to the Ratio Test, for the series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \).

Step 3: Calculate the Ratio

For our series, each term is \( a_n = \frac{1}{n^n} \). Thus, \( a_{n+1} = \frac{1}{(n+1)^{n+1}} \). We calculate:\[\frac{a_{n+1}}{a_n} = \frac{1}{(n+1)^{n+1}} \times \frac{n^n}{1} = \left( \frac{n}{n+1} \right)^n \times \frac{1}{n+1}\]

Step 4: Simplify the Expression

Simplifying the ratio, we have:\[\left( \frac{n}{n+1} \right)^n \times \frac{1}{n+1}\]As \( n \to \infty \), \( \left( \frac{n}{n+1} \right)^n \to \frac{1}{e} \) since it is a known limit, and \( \frac{1}{n+1} \to 0 \).

Step 5: Apply the Limit

Taking the limit as \( n \to \infty \):\[\lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right) = \lim_{n \to \infty} \left( \left( \frac{n}{n+1} \right)^n \times \frac{1}{n+1} \right) = \frac{1}{e} \times 0 = 0\]Since \( 0 < 1 \), according to the Ratio Test, the series converges.

Key Concepts

Ratio Test

The Ratio Test is a valuable tool for determining the convergence of infinite series, particularly when the series involves factorial or exponential terms. For any series \( \sum a_n \), the Ratio Test requires us to evaluate the limit:

• Calculate the ratio \( \frac{a_{n+1}}{a_n} \) for each term.
• Take the limit as \( n \to \infty \): \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \).
• If \( L < 1 \), the series converges.
• If \( L > 1 \) or \( L = \infty \), the series diverges.
• If \( L = 1 \), the test is inconclusive, and another test is needed.

In the given problem, using the Ratio Test on the series \( \sum_{n = 1}^{\infty} \frac{1}{n^n} \), we evaluated the limit and found it to be \( 0 \), indicating convergence as this value is less than 1.

Limit of a Sequence

Understanding the limit of a sequence is crucial when analyzing series convergence. A sequence \( \{a_n\} \) converges to a limit \( L \) if, as \( n \) gets larger, \( a_n \) approaches \( L \). Mathematically, this is written as \( \lim_{n \to \infty} a_n = L \).

• If a sequence converges to zero, it is often useful for series convergence tests.
• For the Ratio Test, the limit \( L \) helps to determine the behavior of the series.
• In our case, we found that \( \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^n = \frac{1}{e} \), which was crucial for solving the problem.

Grasping limits help in broader mathematical contexts too, such as calculus and analysis.

Infinite Series

An infinite series is the sum of an infinite sequence: \( \sum_{n=1}^{\infty} a_n \). These series can sometimes approach a finite limit, leading to convergence, or they can grow without bound, leading to divergence.

• Convergent series have a value, meaning the partial sums \( S_n = a_1 + a_2 + ... + a_n \) approach a specific number as \( n \to \infty \).
• Divergent series either increase to infinity or fail to approach any particular value.
• Understanding convergence helps in various fields, including physics and engineering, where predicting behavior is essential.

The series \( \sum_{n = 1}^{\infty} \frac{1}{n^n} \) is an example of a convergent infinite series, which we proved using the Ratio Test.

Convergence and Divergence

Convergence and divergence are fundamental concepts when working with sequences and series.

• Convergence indicates that the series approaches a specific value, often simplifying complex calculations and predictions. This is especially important in mathematical modeling and simulations.
• Divergence, on the other hand, suggests that the series does not settle to a single value, either oscillating indefinitely or growing without bound.
• These properties can be assessed using various tests, where the Ratio Test is among the most popular for series with factorial or exponential terms.

In the original exercise, convergence was established, telling us that the infinite sum \( \sum_{n = 1}^{\infty} \frac{1}{n^n} \) settles at a finite value, highlighting its predictability and stability for analytical purposes.