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An alternating series is a sequence where the signs of the terms alternate between positive and negative. This usually occurs by raising -1 to an odd or even power. In our sequence formula, \((-1)^n\), the value of \(n\) determines the sign of each term.

• If \(n\) is odd, then the term is negative because \((-1)^{ ext{odd}} = -1\).
• If \(n\) is even, then the term is positive because \((-1)^{ ext{even}} = 1\).

This alternating characteristic is essential when dealing with sequences as it affects the direction or behavior of the series. For example, the first term \(a_1\) in our sequence is -1 because \(n=1\) is odd. The next term \(a_2\) is positive because \(n=2\) is even. This pattern repeats, ensuring a compelling back-and-forth shift in signs.Understanding how alternating series work makes it easier to predict and calculate further terms in any sequence.

The sequence formula is the rule or expression that defines the terms of a sequence.

• For the given exercise, the sequence formula is \(a_{n} = \frac{(-1)^{n}}{n^2}\).
• This formula is a guide to determine the value of each term (\(a_n\)) based on its position in the sequence \(n\).

By using this formula, you can find any term in the sequence by simply plugging in the appropriate value for \(n\). Each \(a_n\) is a rational number that changes in both sign and magnitude depending on \(n\). The formula's alternating sign element \((-1)^n\) defines whether the term is positive or negative, while dividing by \(n^2\) defines how the magnitude changes non-linearly as \(n\) increases. Understanding the sequence formula is crucial for term calculations and determining patterns within the sequence.

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. For example, \(n^2\) means "\(n\) multiplied by itself" (i.e., \(n \times n \)).In the sequence formula \(a_{n} = \frac{(-1)^{n}}{n^2}\), exponents perform two key functions:

• They affect the sign of the term: \((-1)^{n}\) powers switch the sign depending on whether \(n\) is odd or even.
• They diminish the magnitude of each term, as the square of \(n\), \(n^2\), grows quickly, causing the term to become very small as \(n\) becomes large.

Understanding how exponents work is essential for manipulating sequences, as they define both the sign and the scale of sequence terms. Exponents simplify the writing and calculation process of such terms, especially in alternating series where frequent sign changes occur.

Term calculation is the process of determining the value of any specific term in a sequence using its sequence formula. In our example, to find a term \(a_n\), you substitute the given \(n\) into the sequence formula \(a_{n} = \frac{(-1)^{n}}{n^2}\).

• For the 1st term, plug \(n=1\), calculating to \(a_1 = \frac{-1}{1} = -1\).
• The 2nd term uses \(n=2\), leading to \(a_2 = \frac{1}{4}\).
• The approach remains constant regardless of \(n\).

This consistency enables easy calculation of any term. The method works similarly for finding the 100th term, just substitute \(n=100\) to get \(a_{100} = \frac{1}{10000}\).The straightforward calculation assists in identifying patterns or trends within the sequence and aids in solving more complex problems in sequences and series.