Problem 36 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{7}(-1)^{i+1} \cdot i^{2}$$

Step-by-Step Solution

Answer

The sum of the series is 28.

Step 1: Understand the Series

We need to find the sum of the series represented by \( \sum_{i=1}^{7}(-1)^{i+1} \cdot i^{2} \). This summation can be expanded as a sequence of alternating sign squares of integers from 1 to 7.

Step 2: Calculate Each Term

For each integer \( i \) from 1 to 7, compute \((-1)^{i+1} \cdot i^{2}\).- For \( i = 1 \): \((-1)^{1+1} \cdot 1^2 = 1\)- For \( i = 2 \): \((-1)^{2+1} \cdot 2^2 = -4\)- For \( i = 3 \): \((-1)^{3+1} \cdot 3^2 = 9\)- For \( i = 4 \): \((-1)^{4+1} \cdot 4^2 = -16\)- For \( i = 5 \): \((-1)^{5+1} \cdot 5^2 = 25\)- For \( i = 6 \): \((-1)^{6+1} \cdot 6^2 = -36\)- For \( i = 7 \): \((-1)^{7+1} \cdot 7^2 = 49\)

Step 3: Add Up All Terms

Now, add up all the computed terms together:\[ 1 + (-4) + 9 + (-16) + 25 + (-36) + 49 = 28 \]

Step 4: Verify the Result

Recheck the calculations in each term and their sum to ensure accuracy. Each sign alternates because of \((-1)^{i+1}\), which causes positive and negative terms to alternatively subtract and add. Our recheck confirms that the sum is \(28\).

Key Concepts

Alternating Series

An alternating series is a sequence where the signs of the terms consistently switch between positive and negative. In mathematics, this is often represented by a factor of \((-1)^n\) in the series expression.

For example, in the series \( \sum_{i=1}^{7}(-1)^{i+1} \cdot i^{2} \), the \((-1)^{i+1}\) determines whether each term will be positive or negative.
- If \(i\) is odd, \((-1)^{i+1}\) results in a positive term because the exponent \(i+1\) is even.
- If \(i\) is even, \((-1)^{i+1}\) gives a negative term since \(i+1\) becomes odd.

This alternating nature can make series calculations interesting, as it introduces a pattern of adding and subtracting values rather than merely summing positive numbers. When calculating such a series, it's crucial to carefully follow the sign pattern, as miscalculating a sign could lead to errors in the final sum.

Integer Squares

Integer squares are simply the product of an integer multiplied by itself. Mathematically, for an integer \(n\), the square is denoted by \(n^2\). This concept is fundamental and appears in many mathematical contexts, including series calculations, geometry, and algebra.

In our exercise, every term in the series involves squaring an integer value, specifically \(i^2\). For instance:

• For \(i=1\), \(1^2 = 1\)
• For \(i=2\), \(2^2 = 4\)
• And so on up to \(i=7\)

These integer squares are the basis for the calculations in the series, determining the magnitude of each term before considering its sign. Knowing how to calculate basics like integer squares quickly helps in effectively managing other more complex operations.

Series Calculation

Series calculation involves summing a sequence of numbers. In our specific example, the task is to compute the sum of an alternating series of integer squares, as described in the exercise.

Here's how to approach calculating the series step-by-step:

• First, calculate each term based on its position \(i\) in the series. This involves squaring the integer and applying the appropriate sign using \((-1)^{i+1}\).
• Then, add all the terms together while carefully following the sign of each term due to its positioning.

For example, the calculations for the series \( \sum_{i=1}^{7}(-1)^{i+1} \cdot i^{2} \) will look like this:

• Calculate each term individually: 1, -4, 9, -16, 25, -36, and 49.
• Add them considering their signs: \(1 + (-4) + 9 + (-16) + 25 + (-36) + 49\).

The sum, after adding these carefully, results in \(28\).
Re-checking each calculation step ensures accuracy, as the alternation between positive and negative values can be easy to miscalculate.