Problem 35 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{e^{n}}{1+e^{2 n}} $$

Step-by-Step Solution

Answer

The series converges, as its terms resemble those of a convergent geometric series.

Step 1: Recognize the form of the series

The series in question is a sum of terms of the form \( \sum_{n=1}^{\infty} \frac{e^{n}}{1+e^{2n}} \). Here, both the numerator and denominator involve exponential functions.

Step 2: Approximate the series terms

Examine \( a_n = \frac{e^n}{1+e^{2n}} \) as \( n \) becomes very large. Notice that as \( n \to \infty \), \( e^{2n} \gg 1 \) so the denominator \( 1 + e^{2n} \approx e^{2n} \). Thus, the terms can be approximated as \( a_n \approx \frac{e^n}{e^{2n}} = e^{-n} \).

Step 3: Recognize the series form

Recognize that \( e^{-n} \) resembles a geometric series term with a common ratio \( r = \frac{1}{e} \), which satisfies \( 0 < r < 1 \).

Step 4: Apply convergence test for geometric series

For a geometric series to converge, the absolute value of the common ratio must be less than 1. As \( e^{-n} \) indeed forms a geometric series with \( r = \frac{1}{e} < 1 \), the given series converges.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term is a fixed multiple, known as the common ratio, of the preceding term. This type of series is quite common in mathematics and is defined by the expression:

• Terms of a geometric series: \( a, ar, ar^2, ar^3, \ldots \)
• The general term: \( ar^n \)
• Where \( a \) is the initial term and \( r \) is the common ratio.

To determine the convergence of a geometric series, we look at the common ratio \( r \). The series converges if the absolute value of \( r \) is less than 1, that is, \( |r| < 1 \). Conversely, if \( |r| \geq 1 \), the series diverges. This simple condition helps us quickly assess whether the infinite sum of the series will settle to a finite value. In the problem at hand, the series approximates to a geometric series with a common ratio of \( \frac{1}{e} \), which is less than 1, confirming its convergence.

Convergence Tests

Convergence tests are vital tools to determine whether a series converges or diverges. There are several tests available, and selecting the appropriate one depends on the series in question. Common tests include:

• Ratio Test: Analyzes the limit of the absolute value of the ratio of consecutive terms. Best used when terms contain factorials or powers.
• Root Test: Examines the \( n^{th} \) root of the absolute value of terms. Useful for series with terms raised to \( n \).
• Comparison Test: Compares the series with another series that is known to converge or diverge.
• Integral Test: Applies when series terms can be related to a corresponding function.

In the exercise, a type of convergence test for geometric series was applied. Recognizing the series' terms as forming a geometric progression allowed using the geometric series convergence principles. Since the ratio \( \frac{1}{e} \) is less than 1, our convergence criteria are met.

Exponential Functions

Exponential functions are a fundamental type of function where a constant base is raised to a variable exponent, commonly expressed as \( f(x) = a^x \). The most utilized base in calculus and many branches of mathematics is Euler's number, \( e \), approximately equal to 2.718. Characteristics of exponential functions include:

• Rapid growth or decay: Depending on the exponent's sign, the function can grow or decay quickly.
• Continuity and differentiability: Exponential functions are smooth and continuous. Calculating their derivative yields another exponential function.
• Inverse function: The logarithmic function, which appears frequently, especially in solving exponential equations.

In our series, the terms involve exponential functions such as \( e^n \), which become incredibly large as \( n \) increases. However, when \( e^n \) appears in both numerator and denominator, simplification often becomes possible. Recognizing these forms and their properties enables us to simplify the series terms, making analysis, like convergence testing, much more manageable.