Problem 48 - Beginning and Intermediate Algebra

Question

Evaluate each series. $$\sum_{i=1}^{6}(-1)^{i} \cdot(i)$$

Step-by-Step Solution

Answer

The sum of the series \(\sum_{i=1}^{6}(-1)^{i} \cdot(i)\) is \(-3\).

Step 1: Identify the formula to evaluate

We are given the formula: \(\sum_{i=1}^{6}(-1)^{i} \cdot(i)\)

Step 2: Evaluate each term in the series

Now, we can substitute the values of i from 1 to 6 into the formula and evaluate each term: 1. \(i = 1\) \((-1)^{1} \cdot(1) = -1\) 2. \(i = 2\) \((-1)^{2} \cdot(2) = 2\) 3. \(i = 3\) \((-1)^{3} \cdot(3) = -3\) 4. \(i = 4\) \((-1)^{4} \cdot(4) = 4\) 5. \(i = 5\) \((-1)^{5} \cdot(5) = -5\) 6. \(i = 6\) \((-1)^{6} \cdot(6) = 6\) The terms of the series are: -1, 2, -3, 4, -5, 6

Step 3: Sum the evaluated terms

Now let's sum up these terms to get the sum of the series: \(-1 + 2 - 3 + 4 - 5 + 6 = 1 - 2 + 3 - 4 + 5 - 6\)

Step 4: Simplify the sum

We can simplify the sum by adding them pairwise: \(1 - 2 = -1\) \(-1 + 3 = 2\) \(2 - 4 = -2\) \(-2 + 5 = 3\) \(3 - 6 = -3\) And the sum of the series is: \(-3\) So, the evaluation of the series \(\sum_{i=1}^{6}(-1)^{i} \cdot(i)\) is \(-3\).

Key Concepts

Summation Notation

Summation notation is a compact and efficient way to express the sum of a sequence of numbers. You might recognize it by the Greek letter sigma, like this: \( \sum \). This notation lays out a series of terms to be added together according to a set pattern. In our exercise, \( \sum_{i=1}^{6}(-1)^{i} \cdot (i) \) is the summation notation given.

• The lower limit, \( i=1 \), is where our series begins.
• The upper limit, \( i=6 \), tells us where the series ends.
• The expression \((-1)^{i} \cdot (i)\) describes what we are summing over the index \( i \).

By understanding the limits and the pattern of the expression, you can evaluate each term and sum them accordingly.

Alternating Series

An alternating series is a series where the signs of the terms alternate between positive and negative. This is evident in the formula \((-1)^{i} \cdot (i)\). Here, the presence of \((-1)^{i}\) causes the multiplication result to switch signs based on whether \(i\) is odd or even.

• For odd \(i\): The term \((-1)^{i}\) equals \(-1\), making the term negative (e.g., \(i = 1\), yielding \(-1 \cdot 1 = -1\)).
• For even \(i\): The term \((-1)^{i}\) equals \(1\), making the entire term positive (e.g., \(i = 2\), yielding \(1 \cdot 2 = 2\)).

Alternating series require careful attention when summing, as the change in signs affects the overall sum calculation.

Arithmetic Operations

Arithmetic operations form the backbone of algebraic manipulations in series evaluation. In this exercise, you're predominantly adding and subtracting values. Here's a breakdown of how arithmetic operations come into play in our series:- **Addition and Subtraction**: After each term is computed using the expression \((-1)^{i} \cdot(i)\), we either add or subtract the values depending on the expression's sign for each step.- **Pairwise Summation**: Pairing terms for easier calculation steps may help, such as \( (1) - (2) + (3) - (4) + (5) - (6) \) rearranged as \((-1+3+5) + (2+4+6)\) can make simplification straightforward.Understanding and practicing these basic operations is critical, as it helps not just in answering the problem, but forming a bridge to more complex algebraic thinking.

Series Simplification

Series simplification is about making sense of complex summation problems by reducing them to simpler terms. Once all the individual terms are calculated, simplifying the series is often the final step. Here’s the approach from the problem:1. **Calculate Each Term**: With each individual term of the series calculated, combine all terms: \(-1 + 2 - 3 + 4 - 5 + 6\).2. **Group Similar Terms**: A useful technique is to group or rearrange terms for clarity: \((-1+3+5) + (2+4+6)\) and simplify incrementally.3. **Totaling the Series**: Add the results of these calculations to arrive at the total sum, which for this problem was \(-3\).Simplification not only assists in solving problems correctly but also enhances efficiency and clarity in understanding series evaluations.