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An alternating series is a sequence of numbers where the signs of the terms alternate. This means the series has a pattern of positive and negative terms. Such a series is commonly represented as \[ \sum_{n=1}^{\infty} (-1)^n a_n \] where \[ (-1)^n \] dictates the sign of each term:

• When \( n \) is odd, \( (-1)^n \) is -1, making the term negative.
• When \( n \) is even, \( (-1)^n \) is 1, making the term positive.

Understanding alternating series is essential as they appear frequently in mathematics. For example, when looking at the sequence with the general term \( a_n = (-1)^n n^2 \), the alternating behavior causes the sequence to switch between negative and positive numbers, starting with a negative number when \( n = 1 \). This pattern continues with each subsequent term, producing a sequence that looks like: -1, 4, -9, 16, -25, and so on. Each term alternates in sign due to \((-1)^n\). This representation highlights the importance of the alternating sign, creating the beautiful fluctuation within the sequence.

The general term in a sequence provides a formula that defines every term in the sequence as a function of its position (usually denoted as \( n \)). It acts as a key to generating any term in the sequence without having to list all the previous terms. For the sequence given by the general term\[ a_n = (-1)^n n^2 \]the formula tells us:

• The position \( n \) is used to determine both the sign and the magnitude of the term.
• \( n^2 \) computes the square of the position, defining the size of each term.
• \((-1)^n\) alters each term's sign depending on whether \( n \) is odd or even. This determines the alternating nature.

By understanding the general term, you can deduce any term's value without recalculating prior terms. This is particularly helpful for longer sequences and in mathematical proofs where only specific terms are necessary. The general term serves as a compact, mathematical representation that illustrates both the pattern and scaling behavior within the sequence.

The substitution method is a straightforward approach to finding the terms of a sequence or a formula. In this method, you replace the variable in an expression or formula with specific numbers to calculate the desired values. In the context of our sequence:

• The formula is \( a_n = (-1)^n n^2 \).
• By substituting \( n \) with integers like 1, 2, 3, 4, and 5, you generate the respective sequence terms.

For example, when \( n = 1 \), substituting into the formula gives:- \( a_1 = (-1)^1 \times 1^2 = -1 \).Next, if you substitute \( n = 2 \):- \( a_2 = (-1)^2 \times 2^2 = 4 \).This substitution method is methodical, i.e., repeat the process for each subsequent \( n \), like \( n = 3 \), \( n = 4 \), and \( n = 5 \), to find all sequence terms. Applying this method rigorously helps ensure you correctly calculate each term in the sequence based on its position value, while also confirming the logic behind the derived terms. It simplifies problem-solving by breaking down complex sequences into manageable steps.