\[\large 1^m - 2^m + 3^m - 4^m + 5^m - \cdots \]

Define the Abel sum \(\sum\limits_{n=0}^\infty a_n \) to be \( \displaystyle \lim_{z\to 1^-} \sum_{n=0}^\infty a_nz^n, \) if that limit exists.

The closed form of the Abel sum of the (divergent) series as shown above can be written as which of the following functions?

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