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Calculus Level 5

1 m − 2 m + 3 m − 4 m + 5 m − ⋯ \large 1^m - 2^m + 3^m - 4^m + 5^m - \cdots

Define the Abel sum ∑ n = 0 ∞ a n \sum\limits_{n=0}^\infty a_n to be lim ⁡ z → 1 − ∑ n = 0 ∞ a n z n , \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?

Hurwitz zeta function Riemann zeta function Polygamma function Polylogarithm

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Aditya Kumar
May 15, 2016

Relevant wiki: Polylogarithm

Sum can be represented as: S = − ∑ k = 1 ∞ ( − 1 ) k k − m = − Li − m ( − 1 ) S=-\sum _{ k=1 }^{ \infty }{ \frac { { \left( -1 \right) }^{ k } }{ { k }^{ -m } } } =-{ \text{Li} }_{ -m }\left( -1 \right) .

See the relevant wiki for more info.

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