The method simplifies radical infinite products using hyperbolic and trigonometric functions, indicating convergence.
In this paper, the author conducts an in-depth study of a radical infinite product of the form \( ∏ₖ₌ₐ∞f(k)^{2⁻ᵏ}=√{f(a)√{f(a+1)√f(a+2)⋯}} \). The convergence of this expression is briefly discussed. A method for simplifying such infinite products is proposed. By exploring cases where \(f(x)\) represents hyperbolic functions, trigonometric functions, and complex-valued functions, the Dobinski identity is reproduced and generalized. Furthermore, leveraging Weierstrass's theorem and special functions, a relationship is established between this form of infinite nested radical and general infinite products.
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Rong Wu (2025) studied this question.
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