Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the indicated sum in sigma notation. $$ 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{100} $$

Step-by-Step Solution

Answer

The sum is \( \sum_{n=1}^{100} \frac{1}{n} \).

Step 1: Identify the Pattern

We first identify the pattern in the sum. The given sequence is 1, \( \frac{1}{2} \), \( \frac{1}{3} \), ..., \( \frac{1}{100} \). Each term is the reciprocal of an integer, starting from 1 up to 100.

Step 2: Define the General Term

The general term for the sequence can be defined as \( a_n = \frac{1}{n} \), where \( n \) is the position of the term in the sequence. Here, \( n \) will vary from 1 to 100.

Step 3: Set Up the Sigma Notation

Using sigma notation, the sum can be expressed as a series where the index \( n \) starts at 1 and ends at 100. Thus, the sigma notation for the sum is \( \sum_{n=1}^{100} \frac{1}{n} \).

Key Concepts

Series

The concept of a *series* is foundational in mathematics, especially when dealing with sums of sequences. A series is essentially the sum of the terms of a sequence. In the exercise provided, you have a finite series because it starts and ends at particular terms. Here, it begins at 1 and ends at 100.

When expressed in sigma notation, the series becomes concise and easier to understand. Sigma notation uses the Greek letter \( \Sigma \) to represent the sum of a series. For our example, the notation \( \sum_{n=1}^{100} \frac{1}{n} \) follows this pattern, indicating you sum up terms from n = 1 to n = 100.

Key points about series include:

• The lower limit of the sigma notation indicates where the sum begins (in this case, at \( n = 1 \)).
• The upper limit shows where the sum ends (in this problem, at \( n = 100 \)).
• The expression after the sigma symbol provides the general term formula for the sequence.

Understanding these aspects of a series helps simplify complex computations in mathematics by leveraging the pattern of the sequence.

Sequence

A *sequence* in mathematics is an ordered list of numbers, often defined by a specific rule or formula. Each number in the sequence is called a term. In our exercise, we have a sequence of numbers starting with 1, \( \frac{1}{2} \), \( \frac{1}{3} \), and so on, up to \( \frac{1}{100} \).

Sequences are crucial as they lay the groundwork for understanding how a series, particularly when associated with sigma notation, is derived. In any sequence:

• The first term is typically denoted by \( a_1 \).
• Each subsequent term is often defined using a rule or formula involving the position number \( n \). For instance, our sequence uses the formula: \( a_n = \frac{1}{n} \).
• The position number \( n \) is known as the index and can start from any integer value, but conventionally starts from 1.

By working with sequences, you develop methods for predicting and summing entire patterns of numbers, which significantly aids in various mathematical computations.

Reciprocal

The term *reciprocal* refers to one divided by a given number. If you have a number like 5, its reciprocal is \( \frac{1}{5} \). In our example, the sequence is composed of reciprocals of integers, starting from 1.

Understanding reciprocals in a sequence helps make sense of many mathematical operations, especially when dealing with fractions and division. In the exercise, each term in the sequence is a reciprocal of its position number (or index \( n \)).

Here's why reciprocals are important to this exercise:

• They transform integers into fractional forms, allowing for more insightful summation using sigma notation.
• Every denominator in the sequence represents the integer integer whole number place of each term, highlighting its uniqueness in the sum.
• Reciprocals are frequently used in calculus and algebra for simplifying expressions and solving equations.

By grasping the concept of reciprocals, you'll not only master this exercise but also set a solid foundation for many other mathematical concepts and tasks.