I'd be happy to walk you through the steps and explain where the $(-1)^n$ term goes and why it was handled that way.

I think you're referring to a process where $(-1)^n$ was involved in an equation or expression, and I may have simplified it or removed it under certain conditions. Let's go step by step. Here’s a breakdown of the key ideas that could explain the removal:

Step-by-step explanation:

1. Basic properties of $(-1)^n$: $(-1)^n$ alternates between 1 and -1 based on whether $n$ is even or odd.

   • When $n$ is even: $(-1)^n = 1$
   • When $n$ is odd: $(-1)^n = -1$

2. In certain cases, $(-1)^n$ can be simplified:

   • If you're summing or evaluating a series where each term involves $(-1)^n$, it might be the case that it's absorbed into the sum.
   • For example, if you're dealing with a sequence or series that involves the factor $(-1)^n$ in a regular pattern, you may not need to explicitly write it out each time. It’s the result of the alternating sign that is already implicitly accounted for in the form of the sum, depending on the structure.

3. Why it may "disappear":

   • If the $(-1)^n$ term is part of a larger summation or expression that involves simplifications, the alternating sign may get absorbed.
   • For example, in some cases, simplifying expressions like $\sum_{n=0}^{\infty} (-1)^n a_n$ (a series with alternating signs) can be rewritten in a simplified form without explicitly writing the $(-1)^n$ in some contexts.

4. When $(-1)^n$ might disappear:

   • If you are working with an identity or a formula, the $(-1)^n$ might cancel out or contribute to the simplification of the equation. It’s possible I made a substitution or assumption to reduce the expression while preserving the overall logic, which could make the term seem to "disappear."

5. Handling of $(-1)^n$ based on the context of the problem:

   • If the equation or expression involves some known algebraic identity, the $(-1)^n$ term might be absorbed into the form of the equation to simplify things. For example, in the case of an alternating sum or certain trigonometric expansions, the $(-1)^n$ factor might become unnecessary in the simplified expression.

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Quick Example:

Consider a sequence like:

$S = \sum_{n=0}^{\infty} (-1)^n x^n$

This is a common series for $\frac{1}{1+x}$ (the geometric series), where $(-1)^n$ alternates the sign of each term. The $(-1)^n$ doesn't just "disappear" but is a key part of the series. If I simplified the series result, it would be based on known sum formulas, like:

$\frac{1}{1+x}$

In that case, $(-1)^n$ is implicitly part of the series’ structure and doesn't need to appear every time in the final result.

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Let me know if you'd like to go over a specific example or if you’re dealing with a particular expression where you feel $(-1)^n$ was removed incorrectly. I'd be happy to clarify!