Summation method for some divergent series
Euler–Boole summation is a method for summing alternating series . The concept is named after Leonhard Euler and George Boole . Boole published this summation method, using Euler's polynomials , but the method itself was likely already known to Euler.[ 1] [ 2]
Euler's polynomials are defined by[ 1]
2
e
x
t
e
t
+
1
=
∑ ∑ -->
n
=
0
∞ ∞ -->
E
n
(
x
)
t
n
n
!
.
{\displaystyle \displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.}
The periodic Euler functions modify these by a sign change depending on the parity of the integer part of
x
{\displaystyle x}
[ 1]
E
~ ~ -->
n
(
x
+
1
)
=
− − -->
E
~ ~ -->
n
(
x
)
and
E
~ ~ -->
n
(
x
)
=
E
n
(
x
)
for
0
<
x
<
1.
{\displaystyle \displaystyle {\widetilde {E}}_{n}(x+1)=-{\widetilde {E}}_{n}(x){\text{ and }}{\widetilde {E}}_{n}(x)=E_{n}(x){\text{ for }}0<x<1.}
The Euler–Boole formula to sum alternating series is
∑ ∑ -->
j
=
a
n
− − -->
1
(
− − -->
1
)
j
f
(
j
+
h
)
=
1
2
∑ ∑ -->
k
=
0
m
− − -->
1
E
k
(
h
)
k
!
(
(
− − -->
1
)
n
− − -->
1
f
(
k
)
(
n
)
+
(
− − -->
1
)
a
f
(
k
)
(
a
)
)
+
1
2
(
m
− − -->
1
)
!
∫ ∫ -->
a
n
f
(
m
)
(
x
)
E
~ ~ -->
m
− − -->
1
(
h
− − -->
x
)
d
x
,
{\displaystyle {\begin{aligned}\displaystyle \sum _{j=a}^{n-1}(-1)^{j}f(j+h)={}&{\frac {1}{2}}\sum _{k=0}^{m-1}{\frac {E_{k}(h)}{k!}}\left((-1)^{n-1}f^{(k)}(n)+(-1)^{a}f^{(k)}(a)\right)\\&{}+{\frac {1}{2(m-1)!}}\int _{a}^{n}f^{(m)}(x){\widetilde {E}}_{m-1}(h-x)\,dx,\end{aligned}}}
where
a
,
m
,
n
∈ ∈ -->
N
,
a
<
n
,
h
∈ ∈ -->
[
0
,
1
]
{\displaystyle a,m,n\in \mathbb {N} ,a<n,h\in [0,1]}
f
(
k
)
{\displaystyle f^{(k)}}
k th derivative.[ 1] [ 2]
References
^ a b c d Borwein, Jonathan M. ; Calkin, Neil J. ; Manna, Dante (2009), "Euler–Boole summation revisited" , American Mathematical Monthly , 116 (5): 387–412, doi :10.4169/193009709X470290 , hdl :1959.13/940107 JSTOR 40391116 , MR 2510837 ^ a b Temme, Nico M. (1996), Special Functions: An Introduction to the Classical Functions of Mathematical Physics , Wiley-Interscience Publications, New York: John Wiley & Sons, Inc., pp. 17–18 , doi :10.1002/9781118032572 , ISBN 0-471-11313-1 MR 1376370
![{\displaystyle a,m,n\in \mathbb {N} ,a<n,h\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf61dc66a1365493d382d3abb1534afcee71a46e)
