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

First, observe the given series: \( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100} \). Each term in this series is a fraction where the numerator is consistently 1, and the denominator is a sequence of consecutive integers starting from 1 up to 100.

Step 2: Define the General Formula

The general term of the series is \( \frac{1}{n} \), where \( n \) represents each integer from 1 to 100.

Step 3: Set the Sigma Notation

Sigma notation is used to denote the sum of a sequence. In this case, we want to write the sum of the fractions from \( n = 1 \) to \( n = 100 \). This can be expressed in sigma notation as: \[ \sum_{n=1}^{100} \frac{1}{n} \]

Step 4: Verify the Sigma Expression

Check the sigma notation matches the original series. The notation \( \sum_{n=1}^{100} \frac{1}{n} \) correctly represents each term in the series from \( n = 1 \) to \( n = 100 \).

Key Concepts

Series Summation

In mathematics, series summation refers to the process of adding up a sequence of numbers—typically following a specific pattern or rule. For example, the series given in the exercise is a classic example of summing up terms systematically. Each term is added to form the entire sum, also known as the series.

Series can be finite (having a set number of terms) or infinite (continuing indefinitely). The given series is finite because it ends after the 100th term. To represent this succinctly, mathematicians often use symbols and notation, such as the sigma notation.

Using a structured approach like sigma notation not only simplifies expressions, but it also provides clarity, especially when dealing with long or complex series. It allows us to communicate the intention to sum a specific sequential series from a start point, say 1, to an endpoint, in this case, 100.

Harmonic Series

The harmonic series is a particular type of series where each term is the reciprocal of an integer. In the harmonic series, the terms follow this pattern:

• First term: 1 (or \(\frac{1}{1}\))
• Second term: \(\frac{1}{2}\)
• Third term: \(\frac{1}{3}\)
• And so forth...

Harmonic series are distinctive because while they grow increasingly slowly, they do not converge to a limit, meaning they are divergent when extended to infinity. For the exercise problem, however, we only consider the first 100 terms, making it a finite version of the harmonic series.

In practical terms, these series can be seen in real-world scenarios like sound waves and optics, offering insight into phenomena where growth or change decreases incrementally.

Mathematical Notation

Mathematical notation is the language used to write down mathematical ideas, concepts, and formulas. It provides a universal shorthand, allowing mathematicians from around the world to communicate complex ideas efficiently.

In this exercise, sigma notation is an essential element of mathematical notation, symbolizing the sum of a sequence. The sigma symbol (\(\sum\)) is used with an expression to denote the terms to be summed. Following the sigma is an expression indicating:

• The index variable (often \(n\))
• Its starting point (\(n = 1\))
• And its endpoint (\(n = 100\))

By mastering mathematical notation such as this, students can manipulate and understand mathematical concepts more easily, and communicate these concepts accurately in their studies and professional work.