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This paper deals with the inverse of tails of Hurwitz zeta function. More precisely, for any positive integer $s\geq2$ and {$0\leq a<1$}, we give an algorithm for finding a simple form of {$f_{s,a}(n)$} such that $$ \lim_{n\rightarrow\infty}\left\{\left(\sum^\infty_{k=n}\frac{1}{(k+a)^s}\right)^{-1}-{f_{s,a}(n)}\right\}=0. $$ We show that {$f_{s,a}(n)$} is a polynomial in $n-a$ of order $s-1$. All coefficients of {$f_{s,a}(n)$} are represented in terms of Bernoulli numbers.
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I. V. Blagouchine / 2016 / Two series expansions for the logarithm of the gamma function in-volving Stirling numbers and containing only rational coecients for certain arguments related to π-1 / J. Math. Anal. Appl. 442(2):404~434
W. Hwang / A reciprocal sum related to the Riemann zeta function at s = 6 / arXiv: 1709.079941v1
H. Ohtsuka / 2008 / On the sum of reciprocal Fibonacci numbers / Fibonacci Quart 46/47(2):153~159
J. Stirling / 1730 / Methodus dierentialis, sive Tractatus de summatione et interpolatione se-rierum innitarum / Gul. Bowyer
L. Xin / 2016 / Some identities related to Riemann zeta-function / J. Inequal. Appl. 2016:6
W. Hwang / A reciprocal sum related to the Riemann zeta function at s = 6 / arXiv: 1709.079941v1
H. Ohtsuka / 2008 / On the sum of reciprocal Fibonacci numbers / Fibonacci Quart 46/47(2):153~159
J. Stirling / 1730 / Methodus dierentialis, sive Tractatus de summatione et interpolatione se-rierum innitarum / Gul. Bowyer
L. Xin / 2016 / Some identities related to Riemann zeta-function / J. Inequal. Appl. 2016:6
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