Problem 2 - Calculus: Early Transcendentals

Question

Explain what it means to say that \( \sum_{n = 1}^{\infty} a_n = 5. \)

Step-by-Step Solution

Answer

The series converges to the limit 5.

Step 1: Understanding the Summation Notation

The notation \( \sum_{n = 1}^{\infty} a_n \) represents an infinite series, which is the sum of an infinite sequence \( a_1, a_2, a_3, \ldots \). This notation implies that each term \( a_n \) is added together starting from \( n = 1 \) and continuing indefinitely.

Step 2: Convergence of the Series

For an infinite series \( \sum_{n = 1}^{\infty} a_n \) to converge, the partial sums \( S_N = a_1 + a_2 + a_3 + \ldots + a_N \) must approach a finite limit as \( N \) approaches infinity. In this case, the limit of the partial sums is given to be 5.

Step 3: Definition of the Limit

Saying \( \sum_{n = 1}^{\infty} a_n = 5 \) signifies that the limit of the partial sums \( S_N \) is equal to 5 as \( N \) approaches infinity. This means that, as you add more terms from the sequence \( a_n \), the total sum moves closer to the number 5 and eventually becomes arbitrarily close to 5.

Key Concepts

Summation Notation

Summation notation is a mathematical way to represent the sum of a sequence of numbers. You will often see it written with the Greek letter sigma (\( \Sigma \)). In the context of infinite series, such as \( \sum_{n = 1}^{\infty} a_n \), it indicates we are summing all the terms in a sequence starting from \( n = 1 \) and continuing indefinitely.
Each individual term of the sequence is denoted by \( a_n \), where \( n \) is the index of the term. The expression \( \sum_{n = 1}^{\infty} a_n \) tells us that we are to add up the values of each \( a_n \) as \( n \) progresses from 1 to infinity. This notation is very compact and powerful, conveying a lot of information in a small package.

• \(n\) is called the index of summation.
• Starting point for \(n\) is typically 1.
• Summation continues indefinitely towards infinity.

Understanding summation notation is essential for working with series and sequences effectively, setting the stage for further exploration into series convergence and partial sums.

Convergence

Convergence refers to the behavior of an infinite series as more terms are added. Specifically, we are interested in whether the sum of the terms approaches a specific finite value. For the series \( \sum_{n = 1}^{\infty} a_n \), we say it converges if, as you add more terms, the total becomes arbitrarily close to some number, which is the limit of the series.
To determine convergence, we look at the partial sums, \( S_N = a_1 + a_2 + \ldots + a_N \), and investigate their behavior as \( N \), the number of terms, increases. If these partial sums approach a particular number, the series is said to converge to that number.

• A convergent series has partial sums \( S_N \) that approach a finite limit.
• If the limit exists, the series is convergent; otherwise, it is divergent.
• The series \( \sum_{n = 1}^{\infty} a_n = 5 \) implies a limit of 5.

Convergence is a pivotal concept because it tells us that, even though we are adding infinitely many terms, the total value doesn’t just keep growing but settles around a describable value.

Partial Sums

Partial sums are an integral part of understanding infinite series and their behavior. A partial sum \( S_N \) pertains to the sum of the first \( N \) terms of a sequence, represented as \( S_N = a_1 + a_2 + \ldots + a_N \).
The idea is to see how this sum behaves as \( N \) gets larger and larger, giving us insight into whether the infinite series converges or not. In simpler terms, partial sums help us predict the end value of a whole series by examining its segments.

• \( S_N \) stands for the sum of the first \( N \) terms.
• Analyzing \( S_N \) helps in determining the convergence of the series.
• Long-term behavior of \( S_N \) indicates whether \( \sum_{n = 1}^{\infty} a_n \) reaches a finite value.

When you see the expression \( \sum_{n = 1}^{\infty} a_n = 5 \), it implies that as you keep adding more terms, \( S_N \) eventually "settles down" near 5. Partial sums are our guiding tool in studying the overarching behavior of infinite series.