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Although no known closed-form expression exists for the Nth partial sum of the p-series, we show that a closed-form expression can at least be found for the integer part of these partial sums, where 0 < p < 1 , for specific sequences of integers N that are dependent on the value of p. The closed-form expressions follow from an application of the upper and lower bounds obtained from the integral test for the partial sums of series with positive terms. [ABSTRACT FROM AUTHOR]

The article focuses on nested square roots of 2. A closed form expression of Rk when x ⩾ 2,. together with a related nested radical, via an application of a hyperbolic cosine identity. This stands in contrast with the use of the half-angle cosine formula to evaluate the function Rk. As the mapping 2 cosh (.) : [0, ∞) ⃗ [2, ∞) is a bijection, there must exist a unique ξ in [0, ∞) such that x = cosh ξ. By applying the fundamental identity cosh 2 ξ - sinh2 ξ = 1, it is found that 2 sinh ξ = √ x2 - 4, since x = 2cosh ξ.