Euler–Boole summation is a method for summing

alternating series In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternatin ...

based on

Euler's polynomials In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur i ...

, which are defined by :

\frac=\sum_^\infty E_n(x)\frac.

The concept is named after

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

and

George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...

. The periodic Euler functions are :

\widetilde E_n(x+1)=-\widetilde E_n(x)\text \widetilde E_n(x)=E_n(x) \text 0 The Euler–Boole formula to sum alternating series is

: \sum_^(-1)^j f(j+h) = \frac\sum_^ \frac \left((-1)^ f^(n)+(-1)^a f^(a)\right) + \frac 1  \int_a^n f^(x)\widetilde E_(h-x) \, dx, where a,m,n\in\N, a,1/math> and f^is the ''k''th derivative.

References

*Jonathan M. Borwein, Neil J. Calkin, Dante Manna: ''Euler–Boole Summation Revisited''. ''The American Mathematical Monthly'', Vol. 116, No. 5 (May, 2009), pp. 387–412
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*Nico M. Temme: ''Special Functions: An Introduction to the Classical Functions of Mathematical Physics''. Wiley, 2011, , pp. 17–18 {{DEFAULTSORT:Euler-Boole summation Mathematical series Summability methods