Abstract
The 2-periodic function, \(W^*\in C^2({\mathbb {R}})\) is constructed in such a way that the sum \(\sum _{k=1}^n(-1)^{k+1}f(k)\) can be efficiently estimated for any \(n\in {\mathbb {N}}\cup \{\infty \} \) and for every \(f\in C^4[1,\infty )\) having \(\int _1^{\infty }\big |f^{(4)}(x)\big |\,\mathrm {d}\,x<\infty \).
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Notes
In this contribution all numerical computations and graphics are performed using Mathematica [10].
The symbol \(\lfloor x\rfloor \) designates the floor of x (for \(x>0\) its integer part): \(\lfloor x\rfloor =\max \{m\in {\mathbb {Z}}\,|\,m\le x\}\).
A function f is considered as 4-tempered if \(\int _1^{\infty }\left| f^{(4)}(x)\right| \,\mathrm {d}\,x<\infty \).
Consequently \(\underset{n\in {\mathbb {N}},\,n\mapsto \infty }{\lim }f(2n-1)=0\).
For \(S_n\) see Example 6.
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Theodore E. Simos.
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Lampret, V. Simple Effective Estimation of a Smooth Alternating Series. Bull. Malays. Math. Sci. Soc. 43, 1343–1356 (2020). https://doi.org/10.1007/s40840-019-00743-7
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DOI: https://doi.org/10.1007/s40840-019-00743-7