Problem 110 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{1 \cdot 3}-\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}-\frac{1}{4 \cdot 6}+\cdot \cdot \cdot-\frac{1}{10 \cdot 12}$$

Step-by-Step Solution

Answer

The series can be expressed in sigma notation as \(\sum_{n=1}^{10} (-1)^{(n+1)} \cdot \frac{1}{n \cdot (n+2)}\). The exact sum of this series, which can be computed using a graphing utility, is the final answer.

Step 1: Express the series in sigma notation

The series has an overall pattern: \(-\frac{1}{n \cdot (n+2)}\). However, the series alternates signs, with the first term being positive. To account for this we will use the term \((-1)^{(n+1)}\). Expressing the series as a sum using sigma notation from \(n =1\) to \(10\), the series can be written as:\(\sum_{n=1}^{10} (-1)^{(n+1)} \cdot \frac{1}{n \cdot (n+2)}\)

Step 2: Calculate the sum using a graphing utility

With the series now expressed in sigma notation, it can be easily input into a graphing utility or calculator that supports sigma notation to find the exact sum. Remember that you're summing ten terms, so the top number of the sum (n) is 10.

Key Concepts

Alternating Series

An alternating series is a sequence of terms that switch between positive and negative signs. This is evident in the series provided, as each term alternates sign. To have an alternating sign, a common method is to multiply each term by \((-1)^n\) or \((-1)^{(n+1)}\). This factor results in positive values when n is odd and negative values when n is even, or vice versa.

• In the given series, the term \((-1)^{(n+1)}\) is used to ensure the first term is positive, as n starts at 1.
• This contributes to the series' alternating nature, which can be crucial in evaluating convergence and performing calculations.

Understanding alternating series is important because it affects both convergence and the strategy for calculating the sum.

Series Sum Calculation

Calculating the sum of a series involves adding all its terms collectively. With this specific series, there's a need to compute the sum using its representation in sigma notation. However, manually evaluating each term can be tedious and time-consuming. That's why mathematical tools or rules, such as a graphing utility, are often used.

• The formula in sigma notation \(\sum_{n=1}^{10} (-1)^{(n+1)} \cdot \frac{1}{n \cdot (n+2)}\) conveniently represents the described process.
• By inputting this series into a calculator, one can efficiently arrive at the sum of its entire duration, counting 10 terms as in our example.

It's critical to know how this summed value reflects the behavior of alternating series, especially in predicting convergence or divergence.

Graphing Utility

A graphing utility is a powerful tool used to compute complex mathematical expressions, such as summing series. These utilities accommodate automatic calculations that show immediate results. Using a graphing utility is particularly helpful for series in sigma notation.

• It quickly handles the summation of series with many terms, leveraging built-in functions that interpret sigma notation.
• By inputting the series' formula \(\sum_{n=1}^{10} (-1)^{(n+1)} \cdot \frac{1}{n \cdot (n+2)}\), the graphing utility will automate the addition of terms without manual intervention.
• This use also minimizes human error in manual calculations, providing an exact and reliable sum.

Furthermore, some graphing utilities even visualize the alternating series, giving a graphical representation that furthers understanding and analysis.