In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837.
Contents
Technique
Let
A
=
∑
n
=
1
∞
a
n
{\displaystyle A=\sum _{n=1}^{\infty }a_{n}}
be an infinite sum whose value we wish to compute, and let
B
=
∑
n
=
1
∞
b
n
{\displaystyle B=\sum _{n=1}^{\infty }b_{n}}
be an infinite sum with comparable terms whose value is known.
If the limit
γ
:=
lim
n
→
∞
a
n
b
n
{\displaystyle \gamma :=\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}}
exists, then
a
n
−
γ
b
n
{\displaystyle a_{n}-\gamma \,b_{n}}
is always also a sequence going to zero and the series given by the difference,
∑
n
=
1
∞
(
a
n
−
γ
b
n
)
{\textstyle \sum _{n=1}^{\infty }(a_{n}-\gamma \,b_{n})}
, converges.
If
γ
≠
0
{\displaystyle \gamma \neq 0}
, this new series differs from the original
∑
n
=
1
∞
a
n
{\textstyle \sum _{n=1}^{\infty }a_{n}}
and, under broad conditions, converges more rapidly.
We may then compute
A
{\displaystyle A}
as
A
=
γ
B
+
∑
n
=
1
∞
(
a
n
−
γ
b
n
)
,
{\displaystyle A=\gamma \,B+\sum _{n=1}^{\infty }(a_{n}-\gamma \,b_{n}),}
where
γ
B
{\displaystyle \gamma B}
is a constant. Where
a
n
≠
0
{\displaystyle a_{n}\neq 0}
, the terms can be written as the product
(
1
−
γ
b
n
/
a
n
)
a
n
{\displaystyle (1-\gamma \,b_{n}/a_{n})\,a_{n}}
.
If
a
n
≠
0
{\displaystyle a_{n}\neq 0}
for all
n
{\displaystyle n}
, the sum is over a component-wise product of two sequences going to zero,
A
=
γ
B
+
∑
n
=
1
∞
(
1
−
γ
b
n
/
a
n
)
a
n
{\displaystyle A=\gamma \,B+\sum _{n=1}^{\infty }(1-\gamma \,b_{n}/a_{n})\,a_{n}}
.
Example
Consider the Leibniz formula for π:
1
−
1
3
+
1
5
−
1
7
+
1
9
−
⋯
=
π
4
.
{\displaystyle 1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,{\frac {1}{9}}\,-\,\cdots \,=\,{\frac {\pi }{4}}.}
We group terms in pairs as
1
−
(
1
3
−
1
5
)
−
(
1
7