Problem 65 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(1+\frac{k}{n}\right)^{n} $$

Step-by-Step Solution

Answer

The series converges if \( e^k \) is smaller than 1 (which means k is a negative integer), diverges if \( e^k \) is greater than 1 (which means k is a positive integer), and the test is inconclusive if \( e^k \) is equal to 1 (which means k is 0).

Step 1: Recognize Euler's Number

First, recognize that \( \left(1+\frac{k}{n}\right)^{n} \) can be assumed to resemble the definition of Euler's number \( e^{k} \) in the limit as n approaches infinity. Recall that the number 'e' can be defined as \( e = \lim_{{n \to \infty}} \left( 1 + \frac{1}{n}\right)^n \), so for \( k \neq 0 \), \( e^{k} = \lim_{{n \to \infty}} \left( 1 + \frac{k}{n}\right)^n \)

Step 2: Use the Root Test

Next, apply the Root Test, which is useful to determine the convergence of a series whose terms can be expressed as something to the power of 'n'. The Root Test says that a series \( \sum a_n \) converges if \( \lim_{{n \to \infty}} \sqrt[n]{|a_n|} < 1 \), diverges if \( \lim_{{n \to \infty}} \sqrt[n]{|a_n|} > 1 \), and is inconclusive if \( \lim_{{n \to \infty}} \sqrt[n]{|a_n|} = 1 \). Let's see if it can be applied to the given series.

Step 3: Apply Root Test to the Series

Applying the Root Test, each term in the series becomes the absolute value of \( \lim_{{n \to \infty}} \sqrt[n]{\left( 1 + \frac{k}{n}\right)^n} \), which is simply the absolute value of \( e^k \). Since the value of 'k' doesn't change, the limit is continuously \( e^k \).

Step 4: Assess the Result of the Test

Assess the result of the root test: if \( e^k \) is smaller than 1, then the series converges, if \( e^k \) is greater than 1, then the series diverges, and if \( e^k \) is equal to 1 (which means k is 0), the test is inconclusive.

Key Concepts

Euler's Number

Euler's number, denoted as \( e \), is a fundamental constant in mathematics, approximately equal to 2.71828. It is the base of the natural logarithm and is deeply involved with the concept of growth and decay in the field of calculus. One of the primary definitions of \( e \) is as a limit:

• \( e = \lim_{{n \to \infty}} \left( 1 + \frac{1}{n}\right)^n \)

This mathematical limit captures the behavior of continuous growth Understanding \( e \) helps when dealing with series and limits in calculus. In the context of the given exercise, we used Euler's number to recognize the series' form \( \left(1+\frac{k}{n}\right)^{n} \), identifying its limit behavior as \( n \to \infty \) as \( e^k \). Recognizing such patterns simplifies the computation of limits and assists in the application of the Root Test.

Root Test

The Root Test is an important tool in determining the convergence or divergence of an infinite series. The idea is to examine the term within the series when raised to the power of \( n \). The test is structured as follows:

• For a series \( \sum a_n \), calculate \( \lim_{{n \to \infty}} \sqrt[n]{|a_n|} \)
• If the result is less than 1, the series converges.
• If more than 1, it diverges.
• If equal to 1, the Root Test is inconclusive.

In the exercise, applying the Root Test to the series involves computing \( \lim_{{n \to \infty}} \sqrt[n]{\left( 1 + \frac{k}{n}\right)^n} \), which simplifies to \( |e^k| \). Therefore, the convergence strongly depends on the exponential value \( e^k \). If this value is less than 1, then the series is convergent. If greater than 1, it is divergent, and if precisely 1, the Root Test doesn't provide a conclusion.

Sequence and Series

Sequences and series are fundamental concepts in calculus and mathematical analysis. A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.

• A sequence \( \{a_n\} \) is a function from the natural numbers to real numbers.
• A series is commonly denoted as \( \sum_{n=1}^{\infty} a_n \), which means adding an infinite number of terms from a sequence.

In diverging or converging series, the behavior of these sums determines whether they approach a specific value (converge) or not (diverge). Calculating convergence is a critical skill, and understanding these underlying sequences helps in grasping the bigger picture. In our exercise, by considering the series behavior in terms of limits and the definition involving Euler's number, this foundational knowledge of sequences and series becomes instrumental.

Limit

The concept of a limit is central to calculus and analysis. It describes the value that a sequence or function approaches as the input approaches some value.

• The notation \( \lim_{{n \to \infty}} a_n = L \) means that as \( n \) becomes infinitely large, \( a_n \) approaches the value \( L \).
• Limits can often be used to define other mathematical constants, like Euler's number \( e \).

In the context of the series we're analyzing, examining the limit \( \left( 1 + \frac{k}{n} \right)^n \) as \( n \to \infty \) determined whether the series converges or diverges. Recognizing these patterns supports the intuition behind convergence tests like the Root Test, allowing students to understand the robustness of limits in determining series behavior. This helps unlock a deeper appreciation for how limits are used in real-world calculus problems.