Question

Using sigma notation, write the following expressions as infinite series. $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots $$

Short answer

The series is \( \sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n} \).

Step-by-step solution

Analyze the Sequence Pattern

Look at the sequence given: \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \). Notice that the terms alternate between positive and negative. Also, the denominators increase by 1 in each subsequent term.

Identify the General Term Formula

Observe the pattern: positive terms have an odd denominator, and negative terms have an even denominator. The general term for the series is given by: \((-1)^{n+1} \cdot \frac{1}{n}\) where \(n\) represents the term number.

Write the Infinite Series Using Sigma Notation

Using the general term formula, express the series as an infinite series. In sigma notation, the series is represented by \( \sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n} \). This captures both the changing signs and the structure of the terms.

Key concepts

Sigma Notation

When you see a long series of terms to sum up, sigma notation is your best friend. It efficiently expresses the addition of many numbers with a compact format.
Sigma notation uses the Greek letter \( \Sigma \), which stands for sum. It allows you to write an entire sequence in a straightforward form:

• The letter \( n \) is often used as the index of summation, indicating that \( n \) changes from the lower bound to the upper bound of the series.
• For an infinite series, the upper bound is replaced with \( \infty \).
• The expression next to the sigma symbol represents the general or governing rule for the terms of the series.

In our exercise, the series \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \) is neatly captured using sigma notation as:\[\sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{1}{n}\]This notation gives you essential information:

• The series alternates sign, captured by \((-1)^{n+1}\).
• The denominators rise incrementally following the sequence \(\frac{1}{n}\).

Alternating Series

An alternating series is a series where the sign of each term flips from positive to negative, or vice versa.
This characteristic can be expressed through multiplication with \((-1)^n\) or a similar factor.

• In the example series, the alternation is shown as \((-1)^{n+1}\), meaning each odd-indexed term is positive and each even-indexed term is negative.
• This design is crucial for distinguishing alternating series from other series.

Alternating series have special convergence properties and need to be tested with specific criteria, such as the Alternating Series Test:

• This test checks whether the absolute values of the terms decrease progressively and approach zero as the index approaches infinity.

In our case, since \( \frac{1}{n} \rightarrow 0\) as \( n \rightarrow \infty \), the alternating series test can assure convergence, provided the sequence indeed decreases.

General Term Formula

The general term formula of a series defines the rule behind its sequence of terms.
It's like an instruction manual that shows exactly how to generate any term in the series without having to rewrite each composition manually.

• For alternating series, the general term often involves a factor like \((-1)^n\) to reflect the change in signs.
• In our series, the general term formula is given by: \((-1)^{n+1} \cdot \frac{1}{n}\).
• This formula tells us that for every term \(n\), the sign is determined by the expression \((-1)^{n+1}\), while the magnitude is \(\frac{1}{n}\).

Understanding this formula is essential because it simplifies how to represent and calculate the terms' behavior, capturing both the numeric relationship and sign changes effectively.
Whether your series is finite or infinite, identifying its general term formula is key to working out sums, understanding term behaviors, and studying convergence.