Question

Write the indicated sum in sigma notation. $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{100} $$

Short answer

Sum in sigma notation: \( \sum_{n=1}^{100} (-1)^{n+1} \cdot \frac{1}{n} \).

Step-by-step solution

Identify the Structure of the Series

Examine the given series: \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots - \frac{1}{100}\). Notice that it is an alternating series, where the signs alternate between positive and negative with each term.

Determine the General Term

Define the general term of the series. Here, each term is \((-1)^{n+1} \cdot \frac{1}{n}\) for \(n\) from 1 to 100, where \((-1)^{n+1}\) ensures the correct alternating signs.

Set Up the Sigma Notation

Sum the identified general terms from \(n=1\) to \(n=100\). This gives the expression in sigma notation as: \[ \sum_{n=1}^{100} (-1)^{n+1} \cdot \frac{1}{n}. \]

Verify the Components

Check that \((-1)^{n+1} \cdot \frac{1}{n}\) correctly reproduces each term. For example, when \(n=1\), the term is \(1\), and when \(n=2\), the term is \(-\frac{1}{2}\), maintaining the pattern of the series.

Key concepts

Alternating Series

An alternating series is a type of mathematical series where the terms alternate in sign. This means that each positive term is followed by a negative term, and vice versa. In the example we considered, the series begins with 1, then \(-\frac{1}{2}\), followed by \(+\frac{1}{3}\), and continues this alternating pattern.

To identify an alternating series, look for the alternating signs in its terms - this is often indicated by a factor such as \((-1)^{n+1}\), which generates a positive sign for odd indices and a negative sign for even indices. Alternating series can converge even when the absolute values of their terms do not tend to zero, so they are significant in calculus and analysis. Understanding these series is crucial because many natural and mathematical phenomena are modeled using them.

General Term

In any sequence or series, the general term is a formula that represents the terms in the series using a variable, typically denoted as \(n\). The general term enables us to find any term in the series by just plugging in the value of \(n\).

In the series example \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots - \frac{1}{100}\), we express each term \(a_n\) using the formula \((-1)^{n+1} \cdot \frac{1}{n}\). Here, \((-1)^{n+1}\) provides the alternating signs and \(\frac{1}{n}\) matches the decreasing sequence of terms.

• For \(n = 1\), the term is \(1\).
• For \(n = 2\), the term changes because \((-1)^{2+1} = -1\), giving us \(-\frac{1}{2}\).
• This pattern continues up to \(n = 100\).

The general term is beneficial for creating sigma notations and analyzing the behavior of series.

Mathematical Series

A mathematical series is a sum of terms that follow a specific sequence. It is expressed using a sigma (\(\Sigma\)) notation, which concisely represents the addition of terms from a sequence. The series from the original exercise can be written in sigma notation:

\[ \sum_{n=1}^{100} (-1)^{n+1} \cdot \frac{1}{n} \]

This notation tells us to sum the terms generated by the general term \((-1)^{n+1} \cdot \frac{1}{n}\) from \(n = 1\) to \(n = 100\). Series can converge or diverge. Convergence indicates that the sum approaches a specific value, while divergence means it grows infinitely. Understanding series is crucial in fields like finance, physics, and computer algorithms, where cumulative quantities need to be calculated.

Sequence

A sequence in mathematics is an ordered list of elements. Each element in a sequence is called a term. Sequences are defined by a rule that determines how to find each element. For our series, the sequence of terms is defined by the formula \((-1)^{n+1} \cdot \frac{1}{n}\), determining how the elements of the sequence change with each increase in \(n\).

The sequence presents how numbers \(1, -\frac{1}{2}, +\frac{1}{3}, -\frac{1}{4}, \, \cdots, -\frac{1}{100}\) are generated. Identifying the pattern within a sequence is essential for forming series, especially when transforming into sigma notation. Sequences can be finite or infinite.

• Finite sequences have a last element, like \(1\) to \(-\frac{1}{100}\).
• Infinite sequences go on indefinitely without ending.

Knowing how to deal with sequences is foundational in mathematics, as they pave the way to understanding more complex topics like calculus and number theory.