Problem 61 - College Algebra with Modeling and Visualization

Question

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{4}(k+1)$$

Step-by-Step Solution

Answer

The sum of the series is 14.

Step 1: Write Out the Series Terms

The expression for each term in the series is given by \(k+1\). We need to calculate this expression for each value of \(k\) from 1 to 4 and list those terms. Substituting the values: - For \(k = 1\), the term is \(1+1 = 2\).- For \(k = 2\), the term is \(2+1 = 3\).- For \(k = 3\), the term is \(3+1 = 4\).- For \(k = 4\), the term is \(4+1 = 5\).

Step 2: List the Series

Now, list the terms we found from Step 1 in sequence: 2, 3, 4, 5. This forms the series that results from the summation notation \( \, \sum_{k=1}^{4}(k+1) \, \).

Step 3: Evaluate the Sum of the Series

Add the terms of the series together to evaluate the sum:- Start with the first two terms: \(2 + 3 = 5\). - Add this result to the next term: \(5 + 4 = 9\).- Finally, add the last term: \(9 + 5 = 14\). Thus, the sum of the series is 14.

Key Concepts

Summation Notation

Summation notation, also known as sigma notation, is a convenient way to express the sum of a sequence of numbers. It is represented by the Greek letter sigma (\( \Sigma \)). This notation is especially helpful when dealing with a large number of terms or complex series.

In the given exercise, the summation notation \( \sum_{k=1}^{4}(k+1) \) tells us to sum up the expression \(k + 1\) for values of \(k\) starting from 1 going up to 4.

• The lower limit (1) indicates where to start.
• The upper limit (4) tells where to stop.
• The expression \(k + 1\) inside the sigma gives the rule for finding each term.

Recognizing and understanding the components of sigma notation allows you to quickly set up and calculate a series.

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference.

In our series \(2, 3, 4, 5\) derived from \(k + 1\), you can see that each number increases by 1 from the previous number. This makes it an arithmetic progression, because:

• The first term is 2.
• The common difference between terms is 1 (since \( 3-2 = 1, 4-3 = 1, \) and \( 5-4 = 1\)).

The series derived using sigma notation simplifies to understanding how each number builds off the last. Recognizing a sequence as arithmetic simplifies prediction and calculation of terms.

Finite Series

A finite series is one that has a definite number of terms. In contrast to an infinite series, which continues indefinitely, a finite series has a clear start and end.

• The series \(2, 3, 4, 5\) derived from \(\sum_{k=1}^{4}(k+1)\) is finite as it includes only terms from \(k=1\) to \(k=4\).
• Finite series allow for simple calculation of the total sum, as you just add up all terms listed.

Evaluating a finite series, as shown in the solution, involves listing out each term and summing them up, resulting in the sum of 14 for our example.