Problem 11 - Calculus

Question

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

Step-by-Step Solution

Answer

\(\sum_{n=1}^{100} \frac{1}{n}\)

Step 1: Identify the Pattern

Examine the series given: \(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100}\). Notice that each term is in the form \(\frac{1}{n}\), where \(n\) starts at 1 and ends at 100.

Step 2: Determine the General Term

From Step 1, observe that the general term of the series can be expressed as \(\frac{1}{n}\), where \(n\) represents the position of the term in the sequence.

Step 3: Define the Index and Limits

For the sum, the index \(n\) starts at 1 and ends at 100. This gives us the complete range over which the terms should be summed, from \(n = 1\) to \(n = 100\).

Step 4: Write in Sigma Notation

Combine the general term and the limits to express the sum in sigma notation: \(\sum_{n=1}^{100} \frac{1}{n}\). This notation represents the sum of the terms from 1 to 100 as the variable \(n\) varies over that range.

Key Concepts

Series

A series, in mathematics, represents the sum of the elements of a sequence. The given exercise is concerned with the sum of a sequence of fractions: \(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{100}\). Here, each number in this list can be termed as a 'term' in the sequence.

When you add all these terms together, you construct what is called a series. A series is very useful in various branches of mathematics, including calculus, where it is used to represent numbers using infinite sequences. The sum of a finite sequence is a finite series, like in this exercise, where it eventually stops at \(\frac{1}{100}\).

• Helps in summing up large sets of numbers
• Finds use in mathematical analysis and calculus

General Term

The term 'general term' refers to a formula that expresses each term in a series based on its position, often denoted by a variable such as \(n\). For the series \(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100}\), the general formula is \(\frac{1}{n}\). This means the value of each term in the sequence depends on its position in the list.

Understanding the general term is vital because it provides a blueprint for finding each term in the series without having to list them out manually.

Here's why:

• Gives the specific component that can calculate any term in the series
• Simplifies the process of identifying patterns within sequences

Index and Limits

The concept of the 'index' in a series refers to the position of a term within the sequence. In our current context, \(n\) is used as the index to indicate the varying position of each term.

Limits define where the index starts and where it ends. For the series \(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100}\), the limits tell us that \(n\) ranges from 1 to 100.

These two criteria are necessary for expressing the summation concisely:

• Index \(n\) describes the position
• Limits (from 1 to 100) show the range that \(n\) covers

This understanding allows for a structured and orderly calculation when you work out the sum.

Mathematical Summation

Mathematical summation entails adding together the terms of a series. In our case, it is expressed using sigma notation as \(\sum_{n=1}^{100} \frac{1}{n}\). Sigma notation provides a compact and elegant way to indicate this sum over a specified range.

Here's how it works:

• The Greek letter \(\Sigma\) represents the process of summing
• The expression immediately following \(\Sigma\) (\(\frac{1}{n}\)) represents the general term of the series
• \(n=1\) to \(n=100\) specifies the limits of the summation

Thus, mathematical summation using sigma notation allows mathematicians to write potentially very long sums in a concise notation, facilitating both understanding and calculation.