Question

Write out the sums. (You do not need to evaluate them.) \(\sum_{j=1}^{4}(-1)^{j}\)

Short answer

Question: Write out the terms of the sum \(\sum_{j=1}^{4}(-1)^{j}\) without actually calculating the value of the sum. Answer: The terms of the sum are given by \(\sum_{j=1}^{4}(-1)^{j} = -1 + 1 - 1 + 1\).

Step-by-step solution

Understand the sum notation

We are given the sum notation \(\sum_{j=1}^{4}(-1)^{j}\). This means that we have to sum the terms of the function \((-1)^{j}\) for j values from 1 to 4.

Write out the terms

We will now enumerate the terms of the sum, inserting the values of j from 1 to 4 into the function \((-1)^{j}\): - For j = 1: \((-1)^{1} = -1\) - For j = 2: \((-1)^{2} = 1\) - For j = 3: \((-1)^{3} = -1\) - For j = 4: \((-1)^{4} = 1\)

Write the sum

Now that we have written out the terms, we can present the whole sum, without evaluating it: \(\sum_{j=1}^{4}(-1)^{j} = -1 + 1 - 1 + 1\)

Key concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics. They are composed of variables, numbers, and operations that, when combined, form a mathematical phrase. For example, \( (-1)^j \) in our exercise is a simple algebraic expression. Here, \( j \) acts as a variable, and \( (-1) \) is a constant number raised to the power of that variable.

Algebraic expressions can represent a wide range of mathematical scenarios. They allow us to generalize patterns and solve problems systematically. In the given problem, the expression \( (-1)^j \) changes its value as \( j \) takes on different values. Operations like addition, subtraction, multiplication, and division can be intertwined with these expressions to form equations or inequalities.

To simplify the manipulation and evaluation of algebraic expressions in summation, it is essential to first understand how each part operates independently. This basic understanding helps when translating a general expression into specific terms, a step that's crucial in solving exercises involving sums.

Alternating Series

An alternating series is a series in which the terms alternate in sign. In our exercise, each term of the series \( (-1)^j \) alternates between -1 and +1 as \( j \) increases from 1 to 4.

Here is what happens in detail:

• When \( j = 1 \, (-1)^1 \) is equal to -1.
• When \( j = 2 \, (-1)^2 \) is equal to 1.
• When \( j = 3 \, (-1)^3 \) is equal to -1.
• When \( j = 4 \, (-1)^4 \) is equal to 1.

This back-and-forth change of signs is the defining characteristic of alternating series.

Alternating series are important because they have a unique nature and convergence properties. In certain contexts, they can be used to approximate functions that exhibit periodic behavior. Understanding how these series work is beneficial when dealing with complex mathematical models and solutions.

Mathematical Notation

Mathematical notation is the language of mathematical expressions and equations. It is an essential skill to learn for anyone who wants to study mathematics comprehensively. The notation \( \sum_{j=1}^{4} \) is a form of mathematical shorthand that tells us to sum the terms generated by an expression for values of the variable \( j \) from 1 to 4.

In summation notation, the Greek letter sigma (\( \Sigma \)) indicates that we are adding a series of numbers. The terms in the series are determined by replacing \( j \) in the expression, \( (-1)^j \), which is written to the right of the sigma. What's above and below the sigma provides limits to the summation, here indicating that \( j \) starts at 1 and ends at 4.

• Lower limit (\