Problem 1 - Calculus Single Variable

Question

In each of Exercises \(1-12,\) use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n}{e^{n}}\)

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{n}{e^{n}} \) converges by the Ratio Test.

Step 1: Identify the General Term of the Series

The given series is \( \sum_{n=1}^{\infty} \frac{n}{e^{n}} \). The general term \( a_n \) of this series is \( a_n = \frac{n}{e^{n}} \).

Step 2: Apply the Ratio Test

The Ratio Test states that for a series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \) and \( L < 1 \), then the series converges absolutely. We need to find \( \frac{a_{n+1}}{a_n} \) for \( a_n = \frac{n}{e^{n}} \).

Step 3: Calculate the Ratio of Consecutive Terms

Compute \( \frac{a_{n+1}}{a_n} = \frac{\frac{n+1}{e^{n+1}}}{\frac{n}{e^{n}}} \). This simplifies to \( \frac{n+1}{n} \cdot \frac{1}{e} = \frac{n+1}{ne} \).

Step 4: Take the Limit as n Approaches Infinity

Evaluate \( \lim_{n \to \infty} \frac{n+1}{ne} = \lim_{n \to \infty} \frac{n+1}{n} \cdot \frac{1}{e} = \frac{1+\frac{1}{n}}{e} \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \), so the limit becomes \( \frac{1}{e} \).

Step 5: Apply the Conclusion of the Ratio Test

Since the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{e} < 1 \), by the Ratio Test, the series \( \sum_{n=1}^{\infty} \frac{n}{e^{n}} \) converges.

Key Concepts

Convergence of Series

When we talk about the convergence of a series, we are determining whether the sum of a sequence of numbers results in a finite value. If a series converges, you can think of it as the numbers getting closer and closer to a specific value as you continue adding terms. This means that as the number of terms increases indefinitely, the series approaches a certain number instead of growing infinitely large or oscillating without settling.
Understanding convergence is essential in calculus because it helps us determine the behavior of series that appear in various mathematical problems. The Ratio Test is a tool that provides a simple way to check for convergence by analyzing the ratio of successive terms in the series:

• The series converges absolutely if the limit of the absolute value of the ratio of terms is less than 1 when evaluated as the number of terms approaches infinity.
• If the limit equals 1 or doesn't exist, the test is inconclusive, meaning it neither confirms convergence nor divergence.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. These types of series occur frequently in calculus and higher mathematics. Examples include the geometric series, harmonic series, and the series specified in the original exercise, \( \sum_{n=1}^{\infty} \frac{n}{e^{n}} \).
Infinite series can either converge to a finite number or diverge, meaning they'll continue to grow without bound. Determining whether an infinite series converges or diverges can sometimes be complex. However, tests like the Ratio Test allow mathematicians and students to gain insights into the behavior of such series without calculating the partial sums indefinitely. Understanding these concepts is vital for solving calculus problems involving limits, integration, and differential equations.
In practice, the convergence or divergence of an infinite series tells us a lot about the stability and predictability of certain processes described by mathematics. For instance:

• Converging series often imply that a process will stabilize or reach an equilibrium.
• Diverging series might suggest that a process could spiral out of control.

Calculus Exercises

Calculus exercises, particularly those involving series like the one given, often build foundational skills in analyzing continuous change. They frequently involve:

• Recognizing patterns and structures in sequences and series.
• Applying specific tests and techniques, like the Ratio Test, to determine properties of series.

Through these exercises, students develop their ability to handle abstract mathematical concepts and solve complex real-world problems.
The given exercise on determining the convergence of the series \( \sum_{n=1}^{\infty} \frac{n}{e^{n}} \) encourages students to practice the Ratio Test. This exercise illustrates how differential calculus and series analysis interlink to model and solve problems.
By mastering these types of exercises, students will be well-equipped to tackle advanced topics in calculus and beyond, such as solving differential equations or calculating integrals. Ultimately, these problems serve as stepping stones that prepare students for applying calculus in technology, physics, engineering, and economics.