Problem 14 - Thomas Calculus

Question

Which of the series in Exercises 13 46 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.) $$ \sum_{n=1}^{\infty} e^{-n} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} e^{-n} \) converges because \( \frac{1}{e} < 1 \).

Step 1: Understand the Series

We have the infinite series \( \sum_{n=1}^{\infty} e^{-n} \). This is a geometric series where the nth term is given by \( a_n = e^{-n} \), and can also be written as \( \sum_{n=1}^{\infty} \left( \frac{1}{e} \right)^n \). Here, the first term is \( a = \frac{1}{e} \) and the common ratio \( r = \frac{1}{e} \).

Step 2: Check the Series Type

Before proceeding with tests, recognize that this is a geometric series. The general form of a geometric series is \( \sum_{n=1}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. We already identified \( a = \frac{1}{e} \) and \( r = \frac{1}{e} \).

Step 3: Apply the Convergence Test for a Geometric Series

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). In this case, \( |r| = \left| \frac{1}{e} \right| = \frac{1}{e} \). Since \( \frac{1}{e} < 1 \), the series converges.

Key Concepts

Convergence Tests

Convergence tests are essential tools to determine whether an infinite series will converge or diverge. One of the most straightforward tests is the geometric series convergence test. This is particularly useful due to its clarity and simplicity. Convergence tests essentially check the behavior of the series as the number of its terms approaches infinity. Geometric series, defined by a constant ratio between consecutive terms, have a simple criterion: they converge if the absolute value of the common ratio is less than one. This condition ensures that the terms get progressively smaller, ultimately tending towards zero.
For example, consider a geometric series with a common ratio of less than one in absolute value; its terms diminish sufficiently quickly, guaranteeing the sum of infinitely many terms remains finite.

Series Convergence

Series convergence is a concept referring to the sum of all terms in an infinite series approaching a finite limit. This happens when the terms of the series decrease and offset one another's growth adequately.Using a convergence test, such as the geometric series test, helps determine if a series converges. The rule specific to geometric series is determined by the common ratio: if the absolute value of this ratio is less than one, the series converges.Consider the series \( \sum_{n=1}^{\infty} e^{-n} \). Rewriting this series shows that it is geometric with the common ratio \( r = \frac{1}{e} \), which is indeed less than one. Consequently, it converges. Geometric series convergence essentially boils down to checking this ratio.

Infinite Series

Infinite series involve summing an unending sequence of terms. Despite their infinite nature, some infinite series can have a finite sum, which we describe as converging.Two key components define an infinite series:

• The terms, which are the individual elements being summed together.
• The rule or pattern, which dictates how each term relates to the next.

In the case of geometric series like \( \sum_{n=1}^{\infty} e^{-n} \), the terms follow a clear pattern defined by a common ratio \( r \). This makes the series easier to analyze, particularly with convergence tests. If we find that this common ratio's absolute value is less than one, as with our example series, we can be confident that the infinite series has a finite sum.

Common Ratio

The common ratio is a central concept in understanding geometric series. It refers to the constant factor by which we multiply each term to get the next one in the series.A geometric series takes the form \( a_n = ar^n \), where \( a \) is the first term and \( r \) is the common ratio. Emphasizing the importance of the common ratio, it ultimately determines the series' convergence properties.In our exercise, the series \( \sum_{n=1}^{\infty} e^{-n} \) can be rewritten with the common ratio \( r = \frac{1}{e} \). Since this common ratio's absolute value is less than one, the series converges. The common ratio's size not only influences convergence but also the speed at which the terms decrease in size, significantly affecting the series' sum behavior.