Problem 54 - Algebra and Trigonometry

Question

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{n=1}^{100} \frac{(-1)^{n}}{n} $$

Step-by-Step Solution

Answer

The sum \( \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \) is approximately -0.6882.

Step 1: Understanding the Sum

The given sum is \( \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \). This means we need to evaluate the expression \( \frac{(-1)^{n}}{n} \) for each \( n \) from 1 to 100 and add them up.

Step 2: Observing the Pattern

Notice the term \( \frac{(-1)^{n}}{n} \) alternates between being positive and negative due to \((-1)^{n}\). For odd \( n \), the term is negative, while for even \( n \), it is positive.

Step 3: Enter the Expression Into a Graphing Calculator

Using a graphing calculator, enter the expression \( \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \). Use the summation tool or program the calculator to evaluate this sum from 1 to 100.

Step 4: Calculate and Interpret the Result

Once entered, the calculator will compute the sum. The result is an approximation of the alternating harmonic series to the 100th term.

Key Concepts

Graphing Calculator

A graphing calculator can be a powerful tool for evaluating complex sums like the alternating harmonic series. These devices are designed to handle a vast array of mathematical calculations, including summations. To use a graphing calculator for this task, follow these steps:

• Locate the summation function on your calculator. This is often represented by a large sigma symbol (\( \Sigma \)).
• Input the formula for the series: \( \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \).
• Ensure you set the limits of the summation to run from \( n = 1 \) to \( n = 100 \).
• Execute the calculation to find the sum. The calculator will perform the necessary calculations and provide a result, which helps you understand the nature of the series.

By using a graphing calculator, you can explore complex series easily and accurately. This tool saves time and ensures precision, especially when working with tedious manual calculations.

Summation Notation

Summation notation, denoted by the Greek letter sigma (\( \Sigma \)), is a convenient and compact way to represent the addition of a sequence of numbers. In the given exercise, summation notation is used to express the series: \[ \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \] Here,

• \( n=1 \) is the starting point of the series, and \( n=100 \) is its endpoint.
• \( \frac{(-1)^{n}}{n} \) is the general term of the series, where each term alternates in sign due to the \( (-1)^{n} \) factor.

Summation notation succinctly summarizes operations that would otherwise be lengthy to describe. For this alternating series, it quickly conveys the concepts of alternating signs and harmonic terms, allowing a clear view of how the sequence progresses. Understanding summation notation is crucial for simplifying and analyzing large and complex series.

Series Evaluation

Evaluating a series involves calculating the sum of its terms. For the alternating harmonic series \( \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \), the evaluation process is simplified with computational tools, such as graphing calculators, which handle the repetitive nature of series calculations. This particular series alternates in sign for each term. Evaluating it manually requires adding each fraction with its respective sign according to \((-1)^{n}\):

• When \( n \) is odd (e.g., \( n=1 \), \( n=3 \)), the term is negative.
• When \( n \) is even (e.g., \( n=2 \), \( n=4 \)), the term is positive.

Using a graphing calculator simplifies this evaluation by allowing you to input the entire expression and calculate the sum with a single operation. Through this approach, students can focus on understanding the pattern and behavior of the series, rather than being bogged down by complex arithmetic.

Harmonic Series

The harmonic series is an important concept in mathematics, characterized by its terms being the reciprocals of natural numbers: \( \frac{1}{n} \). In the specific case of an alternating harmonic series, such as our exercise, there is an additional factor that causes the terms to alternate in sign. This alternating series is represented by: \[ \sum_{n=1}^{100} \frac{(-1)^{n}}{n} \] Here are some key characteristics of the harmonic series in its alternating form:

• The series never converges to a fixed number. However, the alternating version does converge, providing an approximation that becomes more precise as you sum more terms.
• Each term decreases in magnitude, as the denominator \( n \) increases, making the series's sum approach a certain value.
• The alternating nature of the series comes from \( (-1)^{n} \), which flips the sign with each successive term.

Understanding harmonic series, especially in their alternating form, is crucial for studying limits, convergence, and the behavior of series over an extended range of terms.