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4! 2! · 1! 1 2! 2 1 1 + 3! 3! = −1/120 + 1/24 − 1/8 + 1/36 + 1/16 = −1/720. 2 + 1 2! 4 (27) 378 V. Kowalenko In comparing (27) with (26), we see that there is not only a different number of contributions required to evaluate a4 , but that the numerical values of the contributions are for the most part different. This demonstrates that the partition method for a power series expansion operates in a different manner from the standard recursion relation for the Bernoulli numbers. In Ref. [1] we presented a general formula for the reciprocal logarithm numbers arising out of the partition method.

Therefore, the formal version of Equivalence (1) is given by ∞ ak t k k=0 = ≡ t , et −1 t , et −1 t 2 /4π 2 > −1, t 2 /4π 2 ≤ −1. (12) The divergent behaviour of the series on the lhs of the above result seems to have been overlooked by a great number of mathematical texts including Refs. [2] and [6], although Gradshteyn et al provide the condition for absolute convergence, viz. |t/2π| < 1, when presenting the generating function for the Bernoulli numbers as No. 621 in Ref. [3]. However, the condition does not appear in earlier editions.

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