Problem 57 - Calculus Early Transcendentals

Question

Write the following power series in summation (sigma) notation. $$1-\frac{x}{2}+\frac{x^{2}}{3}-\frac{x^{3}}{4}+\cdots$$

Step-by-Step Solution

Answer

Question: Write the given power series in summation notation: $$1-\frac{x}{2}+\frac{x^{2}}{3}-\frac{x^{3}}{4}+\cdots$$ Answer: The given power series can be written in summation notation as $$\sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{x^{n-1}}{n}$$

Step 1: Identify the pattern

Observe the given power series: $$1-\frac{x}{2}+\frac{x^{2}}{3}-\frac{x^{3}}{4}+\cdots$$ The pattern that we notice in the series is: 1. The sign alternates between positive and negative for each new term. 2. The exponent of x increases by 1 each time. 3. The denominator of the fraction increases by 1 each time.

Step 2: Determine the general term

Make use of the observations to come up with a general term for the power series. We can write the general term as: $$(-1)^{n}\cdot\frac{x^{n-1}}{n}$$ This general term takes care of the alternating signs (using \((-1)^{n}\)), the increasing exponent of x (using \(x^{n-1}\)), and the increasing denominator (using \(n\)).

Step 3: Write the series in sigma notation

Now that we have the general term, we can write the power series in sigma (summation) notation. Since the series starts at n = 1, we have: $$\sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{x^{n-1}}{n}$$ This is the required sigma notation for the given power series: $$1-\frac{x}{2}+\frac{x^{2}}{3}-\frac{x^{3}}{4}+\cdots = \sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{x^{n-1}}{n}$$

Key Concepts

Sigma Notation

Sigma notation is a short and concise way of expressing a long sum of terms. This is highly useful when working with power series, as it allows us to represent infinite series simply and clearly. In sigma notation, the Greek letter \( \Sigma \) is used, which represents the word 'sum.' The terms to be added are placed right of the sigma symbol, and underneath it, you'll often see an expression indicating the index range, usually in the form \( n=a \). This shows where to start, and above \( \Sigma \), it shows where to finish or indicates infinity if the series is never-ending.

Consider the series presented in the original exercise:

• The series is infinite, as indicated by the successive ellipses.
• Sigma notation lets us compress this infinite sequence using the general term derived earlier.
• The expression \( \sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{x^{n-1}}{n} \) represents our series in a neat and standardized way.

Using sigma notation not only saves time but also helps when analyzing the behavior of the series, its convergence, and understanding its overall pattern.

General Term

The general term in a series is a crucial component as it defines how each term in the series is constructed. To determine a series in mathematical terms, you must first understand its pattern and behavior.

For the given power series:

• The general term here, \((-1)^{n} \cdot \frac{x^{n-1}}{n}\), compacts the pattern into a single formula.
• Each component of the term captures a unique attribute of the series' behavior:
  • \((-1)^{n}\) switches the sign of each term, capturing the alternation between positive and negative.
  • The \(x^{n-1}\) expression maps how the power of \(x\) grows with each term.
  • The denominator \(n\) represents the rising numeric sequence within each fraction.

Determining the general term is essential for converting a series into sigma notation, aiding in easier computations and insights.

Alternating Series

An alternating series is a type of series where the terms alternate in sign. This type of behavior can be spotted through the presence of a \((-1)^{n}\) or \((-1)^{n+1}\) in the general term. This ensures that each term in the series flips between positive and negative, creating an oscillating pattern.

In our example,

• The presence of \((-1)^{n}\) in the general term \((-1)^{n} \cdot \frac{x^{n-1}}{n}\) creates this alternating behavior.
• Each added term will change sign compared to the previous one, leading to a pattern of addition and subtraction.
• Alternating series are interesting because they exhibit specific converging properties that are not always present in other types of series.
  • For example, the Alternating Series Test provides criteria under which these series converge.

Understanding alternating series can be crucial for tackling problems involving series convergence and for identifying the nature of series in systematic ways.