Problem 62 - Beginning and Intermediate Algebra

Question

Write each series using summation notation. -1+4-9+16-25

Step-by-Step Solution

Answer

\[\sum_{n=1}^{5}(-1)^{(n+1)}n^2\]

Step 1: Finding the General Term

Since the absolute values of the terms are perfect squares, let us denote this by n^2, where n is an integer. Now, to account for the alternating signs, we can use (-1)^m, where m is an integer. In this case, we need to find the correct value of m such that when multiplied by (-1), we get the alternating signs in the series.

Step 2: Determine the Correct Power for the Alternating Sign

In our series, the first term is negative, so when n is equal to 1, we want (-1)^m to be negative. We can achieve this by making m equal to odd numbers. When n is odd, we want (-1)^m to be negative, and when n is even, we want (-1)^m to be positive. We can do this by using the expression (-1)^(n+1) for the alternating sign: - When n=1 (odd), the expression becomes (-1)^(1+1) = (-1)^2 = 1 - When n=2 (even), the expression becomes (-1)^(2+1) = (-1)^3 = -1 The alternating sign expression for this series is, therefore, (-1)^(n+1). So, the general term with alternating signs and perfect squares is (-1)^(n+1)n^2.

Step 3: Writing the Series using Summation Notation

Now that we have the general term (-1)^(n+1)n^2, we can represent the series using summation notation. As we have only 5 terms in the series, we will be summing from n=1 to n=5. So, the summation notation for this series is: \[\sum_{n=1}^{5}(-1)^{(n+1)}n^2\] Now you have rewritten the given series using the summation notation.

Key Concepts

Alternating Series

An alternating series is characterized by its terms switching signs between positive and negative as you move from one term to the next. This alternating pattern is captured mathematically using the expression \((-1)^n\) or \((-1)^{n+1}\). In these formulas:

• If \(n\) is an odd number, \((-1)^n\) gives us a negative term.
• If \(n\) is an even number, \((-1)^n\) provides a positive term.

• The formula \((-1)^{n+1}\) flips this pattern.

Such series are common in mathematics as they help describe various natural patterns and processes. For the given series, \(-1 + 4 - 9 + 16 - 25\), the use of \((-1)^{n+1}\) effectively creates the desired alternating signs.

General Term

The general term in a series is a formula that allows us to find any term in the sequence by plugging in the term's position number. For a series that involves alternating signs and perfect squares, like our example series \(-1 + 4 - 9 + 16 - 25\), the general term can be expressed as:\[(-1)^{(n+1)}n^2\]
Here:

• \((-1)^{(n+1)}\) manages the alternating signs.
• \(n^2\) refers to the perfect squares.

This formula is designed such that when you input the position number \(n\), it will output the correct term in the sequence.

Perfect Squares

Perfect squares are numbers that are the square of an integer. In the context of a series, they help in identifying a pattern or rule that can be uniformly applied. Consider a few examples to illustrate:

• For example, \(1\times1 = 1\), \(2\times2 = 4\), \(3\times3 = 9\), and so on.

In our series, \(-1 + 4 - 9 + 16 - 25\), each term in absolute value corresponds to a perfect square (\(1, 4, 9, 16, 25\)). Perfect squares such as these provide a predictable element, forming the basis for constructing each term in the series.

Series Notation

Series notation, specifically summation notation, is a concise way to represent a series or sequence of numbers using symbols and indices. It is expressed using the summation symbol \(\Sigma\) followed by the terms of the series. To express the series \(-1 + 4 - 9 + 16 - 25\) in summation notation:\[\sum_{n=1}^{5}(-1)^{(n+1)}n^2\]Where:

• \(n=1\) to \(n=5\) indicates the starting and ending indices of the series.
• \((-1)^{(n+1)}n^2\) is the general term of the series.

This notation efficiently captures the entire series in a single expression, making it easy to comprehend and communicate. Understanding series notation is essential for working with complex series and sums across mathematics.