Problem 42 - Precalculus: Mathematical for Calculus

Question

Find the sum. $$\sum_{j=1}^{100}(-1)^{j}$$

Step-by-Step Solution

Answer

The sum alternates between -1 and 1, totaling zero.

Step 1: Understanding the Summation

The expression given is \( \sum_{j=1}^{100}(-1)^{j} \). This represents a summation where each term is \((-1)^{j}\) and the index \(j\) runs from 1 to 100. Our task is to calculate the sum of these terms.

Step 2: Evaluating the Expression Within the Sum

The expression \((-1)^{j} \) takes on values based on whether \( j \) is odd or even. If \( j \) is odd, \((-1)^{j} = -1\). If \( j \) is even, \((-1)^{j} = 1\).

Key Concepts

Alternating Series

An alternating series is a series where the terms switch signs between positive and negative as you move from one term to the next. This typically happens when each term is multiplied by (-1) raised to the power of the term's position. In a formula like \\( \sum_{j=1}^{n} (-1)^{j} a_j \), the sign of each term \( a_j \) alternates based on whether \( j \) is even or odd. For our example, \((-1)^j\) becomes - 1 if \( j \) is odd, meaning the term is negative.- 1 if \( j \) is even, meaning the term is positive.A simple alternating series can look like: -1, 1, -1, 1, ..., etc.With the series \( \sum_{j=1}^{100} (-1)^j \), we see an equal number of positive and negative terms (since 100 is even). Hence, they can perfectly cancel each other, leading to a sum of zero.

Mathematical Series

A mathematical series is the sum of the terms of a sequence. It can involve a finite number of elements, like in our case, or continue indefinitely. The series is presented in a summation notation which is easier and more compact to work with.
To sum a series effectively, understanding the pattern of the terms is crucial. If the sequence has certain repeating characteristics, it simplifies calculating the sum. For alternating series, like our exercise, recognizing that the terms with alternating signs can lead to cancellations

• Odd-indexed terms contribute a negative value
• Even-indexed terms contribute a positive value
• The series ends when all terms have been accounted for

This understanding can make the operation much more straightforward.

Even and Odd Functions

Even and odd functions attach specific properties to functions based on their symmetry.An **even function** satisfies \( f(x) = f(-x) \) for any \( x \), meaning it is symmetrical about the y-axis. For instance, cosine is an even function: \( \cos(x) = \cos(-x) \).
An **odd function** satisfies \( f(-x) = -f(x) \), meaning it is symmetrical about the origin, and reflects over the origin. Sine is an example of an odd function: \( \sin(-x) = -\sin(x) \).
While the terms \( (-1)^j \) from the series are not even or odd functions, the concept relates to how terms change with each increment of \( j \). Odd position terms like \((-1)^1, (-1)^3\) naturally take on \(-1\), whereas even position terms like \((-1)^2, (-1)^4\) resolve to \(1\). Understanding this behavior is crucial in quickly recognizing the pattern in alternating series.