A Constructed Complex Identity Involving Euler's Number and a Rapidly Convergent Infinite Series

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Research Paper (postgraduate) from the year 2026 in the subject Physics - Applied physics, grade: A, , language: English, abstract: Euler¿s number e is one of the central constants in mathematics, appearing in analysis, differential equations, probability, and complex variables. In this paper, we examine the identity e = 4th root of 54 + [((-1)^(-i/¿) ¿ 4th root of 54) / ¿ from k = 1 to ¿ of V5 / (¿k)^(5k)] × ¿ from k = 1 to ¿ of V5 / (¿k)^(5k) The expression combines a complex power, a rapidly convergent infinite series, and a fourth root of an integer. Using the principal branch of the complex logarithm, the term (-1)^(-i/¿) is evaluated exactly as e. The infinite series ¿ from k = 1 to ¿ of V5 / (¿k)^(5k) is absolutely convergent and decays extremely fast because the exponent grows with k. After substituting the complex term and simplifying algebraically, the expression reduces to 4th root of 54 + (e ¿ 4th root of 54) = e Thus, the identity is verified exactly. Although the formula does not define a new constant or provide a new representation of e, it serves as an instructive example of how complex exponentiation and infinite series can be combined into a compact symbolic identity. The paper highlights the role of branch selection in complex analysis, the behavior of rapidly convergent series, and the algebraic cancellation that leads to the final result.

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