General Mathematics

    

The Antiderivative Sequence and Its Series: the Rediscovery of the Polylogarithm from Fundamental Calculus

Authors: Marciano L. Legarde

This study explores the Antiderivative Power Rule Sequence, demonstrating how its infinite series leads to the polylogarithm. By iteratively applying the power rule for antiderivatives to successive powers of x, we derive the sequence, which, when expressed as an infinite series, converges to -ln(1-x). Differentiating the resulting series recovers the geometric series, highlighting a profound inverse relationship between 1/(1-x) and -ln(1-x). Furthermore, this formulation establishes a natural connection to the polylogarithm function, generalizing the relationship for higher orders of integration. This work provides both pedagogical and theoretical insights, reconstructing a transcendental function from elementary calculus operations.

Comments: 7 Pages.

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[v1] 2025-10-14 08:46:26

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