1 − 1 + 2 − 6 + 24 − 120 + ⋯

ID: 1-minus-1-plus-2-minus-6-plus-24-minus-120-plus-⋯

To evaluate the series \( S = 1 - 1 + 2 - 6 + 24 - 120 + \cdots \), we can identify the terms in the series in a more systematic way. We observe that the series can be expressed in terms of factorials: - The \( n \)-th term appears to follow the pattern \( (-1)^n n! \).

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