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.
Euler's polynomials are defined by
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}
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 }
f
(
k
)
{\displaystyle f^{(k)}}
k th derivative.
References
^ 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
^ 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
Categories :
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
**DISCLAIMER** We are not affiliated with Wikipedia, and Cloudflare.
The information presented on this site is for general informational purposes only and does not constitute medical advice.
You should always have a personal consultation with a healthcare professional before making changes to your diet, medication, or exercise routine.
AI helps with the correspondence in our chat.
We participate in an affiliate program. If you buy something through a link, we may earn a commission 💕
↑
:
and 
is the kth derivative.
