Problem 66 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} e^{-n} $$

Step-by-Step Solution

Answer

The given series converges because the absolute value of the ratio is less than 1.

Step 1: Identify The Series Type

The series given is \(\sum_{n=1}^{\infty} e^{-n}\), which is an infinite geometric series with common ratio \(r = e^{-1}\). We want to determine whether this series converges or diverges.

Step 2: Apply The Basic Convergence Test

The basic convergence test for a geometric series can be applied here. The test states that an infinite geometric series \(\sum_{n=1}^{\infty} a r^{n-1}\) is convergent if and only if \(|r| < 1\). Considering our series, the ratio, \(r = e^{-1} = 1/e\). Hence, \(|r| = |1/e| = 1/e\).

Step 3: Test The Convergence Condition

Since the absolute value of the ratio, \(|1/e|\), is less than 1, using the basic convergence test, we conclude that the series converges.

Key Concepts

Understanding Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This special form of series is prevalent in mathematics due to its simple structure and the ease with which we can determine its convergence.

In the given exercise, you were faced with an infinite geometric series of the form \(\sum_{n=1}^{\infty} a r^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio. Geometric series can be used to model many real-world scenarios, such as calculating the total amount of a loan or investment over time, given a constant rate of interest.

For the series to converge, meaning the sum approaches a finite value as we add more and more terms, the common ratio has to be strictly between -1 and 1. If \(r\) falls outside this range, the series will either grow infinitely large or oscillate without settling on any one value, hence diverging.

Convergence Tests for Series

When working with series, it's crucial to determine whether they converge or diverge. To do that, convergence tests are employed. These tests apply specific criteria to decide if the sum of the infinite series reaches a finite limit.

The basic convergence test for a geometric series requires the absolute value of the common ratio \(r\), denoted as \(\|r\|\), to be less than one, \(\|r\| < 1\). The series presented in the original exercise, \(\sum_{n=1}^\infty e^{-n}\), has a common ratio of \(e^{-1}\) which is approximately 0.3679. Since this value is less than 1, we can confidently say that our series passes the geometric series convergence test and thus, converges.

There are many other convergence tests like the Ratio Test, Root Test, and Integral Test, which apply to series that have different attributes than a geometric series. Choosing the right test is vital, as it simplifies the process and ensures a correct conclusion.

The Exponential Function in Series

The exponential function, commonly represented by \(e^x\), is a mathematical function that appears frequently in various realms, such as compound interest, population growth, and, in the context of our original exercise, infinite series. Its reciprocal \(e^{-x}\) is used to describe decay processes.

In our case, \(e^{-n}\) represents the terms of the series. This decay factor causes each subsequent term of the series to be smaller than the preceding one, which is typical behaviour in applications dealing with decay, such as radioactive decay or depreciation of value over time.

Understanding the behaviour of the exponential function is key to solving problems related to growth and decay. It exemplifies how even a simple mathematical concept can be profoundly applied to describe the world around us—whether in finance, biology, physics, or countless other fields where exponential changes occur.