Problem 35 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{k=1}^{6}(-1)^{k} \cdot k$$

Step-by-Step Solution

Answer

The sum of the series is 3.

Step 1: Understand the Series

We are given the series \( \sum_{k=1}^{6}(-1)^{k} \cdot k \). This notation indicates we need to sum the terms generated by the expression \((-1)^{k} \cdot k\) as \(k\) varies from 1 to 6.

Step 2: Calculate Each Term in the Series

Calculate each term of the series using the expression \((-1)^{k} \cdot k\):- For \(k = 1\): \((-1)^{1} \cdot 1 = -1\)- For \(k = 2\): \((-1)^{2} \cdot 2 = 2\)- For \(k = 3\): \((-1)^{3} \cdot 3 = -3\)- For \(k = 4\): \((-1)^{4} \cdot 4 = 4\)- For \(k = 5\): \((-1)^{5} \cdot 5 = -5\)- For \(k = 6\): \((-1)^{6} \cdot 6 = 6\)

Step 3: Sum the Calculated Terms

Now, sum up all the terms calculated:- \(-1 + 2 - 3 + 4 - 5 + 6\).Let's group and sequentially add these:1. \(-1 + 2 = 1\)2. \(1 - 3 = -2\)3. \(-2 + 4 = 2\)4. \(2 - 5 = -3\)5. \(-3 + 6 = 3\)

Step 4: Conclude with the Final Sum

The sum of the series \( \sum_{k=1}^{6}(-1)^{k} \cdot k \) is 3. We obtained this by sequentially summing all individual terms calculated in the previous steps.

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. This notation makes calculations for series much more organized and easier to comprehend. The symbol for summation is the Greek capital letter sigma, denoted as \( \Sigma \). When using summation notation, you typically see it in the form of \( \sum_{k=a}^{b} f(k) \). Here:

• \( k \) is the index of summation, which runs from \( a \) to \( b \).
• \( f(k) \) is the expression that defines the terms in the series.
• \( a \) is the lower limit where the summation starts, and \( b \) is the upper limit where it ends.

In our given series, \( \sum_{k=1}^{6}(-1)^{k} \cdot k \), it shows that you need to calculate the values of \((-1)^{k} \cdot k\) for each integer value of \( k \) starting from 1 and ending at 6, then sum those values together. Summation notation can greatly simplify discussions about series by replacing lengthy expressions with clear, concise symbols.

Alternating Series

An alternating series is a series whose terms alternate in sign. The series often takes the form of a sequence like \( a_1 - a_2 + a_3 - a_4 + \ldots \), where each term switches from positive to negative and vice versa. This behavior is crucial for understanding and predicting the summation result.

• In our series, \((-1)^k\) is responsible for creating the alternating pattern. When \(k\) is odd, \((-1)^k\) yields \(-1\), resulting in negative terms.
• Conversely, when \(k\) is even, \((-1)^k\) results in \(1\), yielding positive terms.

This alternation helps balance sums and sometimes simplifies the calculation by cancelling out terms to some extent. Alternating series are seen in many mathematical contexts, and their alternating nature can often lead to interesting summation results.

Algebraic Manipulation

Algebraic manipulation involves rearranging and combining terms to simplify the calculation of a series. It's a vital skill when working with series and can especially be useful in alternating series where things seem to cancel out.For the series \( \sum_{k=1}^{6}(-1)^{k} \cdot k \), the process involved calculating and summing each term explicitly:

• The first step calculated each term: \(-1, 2, -3, 4, -5,\) and \(6\).
• Then, we grouped them to assist in effective combination: \((-1 + 2), (-3 + 4), (-5 + 6)\).

Each group simplifies to \(1, 1,\) and \(1\), where summing gives us a total of 3. By understanding the interplay between terms due to their alternating signs, you effectively manipulate and simplify the addition, leading to the final result. This technique not only makes calculations easier but often helps in detecting patterns or shortcuts in problem-solving.