Problem 46 - Beginning and Intermediate Algebra

Question

Evaluate each series. $$\sum_{i=1}^{3} \frac{2 i}{i+3}$$

Step-by-Step Solution

Answer

The sum of the given finite series, \( \sum_{i=1}^{3} \frac{2 i}{i+3} \), is \(\frac{23}{10}\).

Step 1: Write down the series

First, let's write down the series with its terms: \( \sum_{i=1}^{3} \frac{2 i}{i+3} = \frac{2(1)}{1+3} + \frac{2(2)}{2+3} + \frac{2(3)}{3+3} \)

Step 2: Calculate each term of the series

Next, we compute each term in the series: Term 1: \( \frac{2(1)}{1+3} = \frac{2}{4} = \frac{1}{2} \) Term 2: \( \frac{2(2)}{2+3} = \frac{4}{5} \) Term 3: \( \frac{2(3)}{3+3} = \frac{6}{6} = 1 \)

Step 3: Sum all terms

Finally, sum all terms together: \( \frac{1}{2} + \frac{4}{5} + 1 \) To add these fractions, we first need to find a common denominator, which in this case is 10. Thus, we rewrite the fractions as: \( \frac{5}{10} + \frac{8}{10} + \frac{10}{10} \) And now, we can easily add them: \( \frac{5+8+10}{10} = \frac{23}{10} \)

Step 4: Write the result

Finally, we can write down our result: \( \sum_{i=1}^{3} \frac{2 i}{i+3} = \frac{23}{10} \)

Key Concepts

Summation Notation

Summation notation is a powerful way to express the sum of a sequence of numbers. Instead of writing out each individual term, we use the Greek letter sigma, \( \Sigma \), followed by a formula that generates each term in the series as the index \( i \) changes. In our original exercise, the expression \( \sum_{i=1}^{3} \frac{2i}{i+3} \) indicates that we need to sum the terms generated by the formula \( \frac{2i}{i+3} \) for every integer \( i \) from 1 to 3. This compact form makes calculations neater and saves space when dealing with larger sequences.

Each element in the summation can be calculated by substituting \( i \) with the integers from the starting point to the endpoint. It's an efficient method to outline calculations and is widely used in mathematics and statistics to represent sums and series clearly.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In our series \( \frac{2i}{i+3} \), the numerator is \( 2i \), a linear polynomial, and the denominator is \( i+3 \), also a linear polynomial. Rational expressions are like regular fractions, but they involve variables that can take on different values according to the expression's rule.

When dealing with rational expressions, especially in summation notation, it's important to carefully substitute the variables and simplify each term correctly. Keep in mind that the basic operations on fractions, like finding a common denominator or reducing terms, are still used but apply to expressions with variables.

Common Denominator

Finding a common denominator is crucial when adding fractions that have different denominators. To add fractions together, you must first rewrite them so that they have the same denominator. In the step provided, fractions \( \frac{1}{2} \), \( \frac{4}{5} \), and \( 1 \) need a common denominator.

To determine a common denominator, look for the least common multiple (LCM) of the denominators—in this case, 2, 5, and 1. The LCM is 10. We then convert each fraction: change \( \frac{1}{2} \) to \( \frac{5}{10} \), \( \frac{4}{5} \) to \( \frac{8}{10} \), and \( 1 \) to \( \frac{10}{10} \).

This process ensures that each fraction is expressed in tens, allowing for straightforward addition. It's a fundamental skill in handling fractions and is especially useful when encountering rational expressions in series.

Fraction Addition

Adding fractions involves lining them up with a common denominator and then combining their numerators. Once the fractions have matching denominators, the numerators can be added directly. For example, once we have \( \frac{5}{10} \), \( \frac{8}{10} \), and \( \frac{10}{10} \), they are easy to add:

• Combine the numerators: \( 5 + 8 + 10 = 23 \)
• Place the sum over the common denominator: \( \frac{23}{10} \)

The result is \( \frac{23}{10} \), indicating the total sum of the series expressed in terms of a single fraction.

Understanding fraction addition is crucial in mathematics, as fractions are commonly encountered in various math problems. Getting comfortable with these operations is key for tackling more complex calculations involving rational expressions and series.