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.
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 − 1 9 ) + ⋯ = 1 − 2 ( 1 15 + 1 63 + ⋯ ) = 1 − 2 A {\displaystyle {\begin{aligned}&1-\left({\frac {1}{3}}-{\frac {1}{5}}\right)-\left({\frac {1}{7}}-{\frac {1}{9}}\right)+\cdots \\&\quad =1-2\left({\frac {1}{15}}+{\frac {1}{63}}+\cdots \right)=1-2A\end{aligned}}}
where we identify
A = ∑ n = 1 ∞ 1 16 n 2 − 1 . {\displaystyle A=\sum _{n=1}^{\infty }{\frac {1}{16n^{2}-1}}.}
We apply Kummer's method to accelerate A {\displaystyle A} , which will give an accelerated sum for computing π = 4 − 8 A {\displaystyle \pi =4-8A} . Let
B = ∑ n = 1 ∞ 1 4 n 2 − 1 = 1 3 + 1 15 + ⋯ = 1 2 − 1 6 + 1 6 − 1 10 + ⋯ {\displaystyle {\begin{aligned}B&=\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}={\frac {1}{3}}+{\frac {1}{15}}+\cdots \\&={\frac {1}{2}}-{\frac {1}{6}}+{\frac {1}{6}}-{\frac {1}{10}}+\cdots \end{aligned}}}
This is a telescoping series with sum value 1⁄2. In this case
γ := lim n → ∞ 1 16 n 2 − 1 1 4 n 2 − 1 = lim n → ∞ 4 n 2 − 1 16 n 2 − 1 = 1 4 {\displaystyle \gamma :=\lim _{n\to \infty }{\frac {\frac {1}{16n^{2}-1}}{\frac {1}{4n^{2}-1}}}=\lim _{n\to \infty }{\frac {4n^{2}-1}{16n^{2}-1}}={\frac {1}{4}}}
and so Kummer's transformation formula above gives
A = 1 4 ⋅ 1 2 + ∑ n = 1 ∞ ( 1 − 1 4 1 4 n 2 − 1 1 16 n 2 − 1 ) 1 16 n 2 − 1 = 1 8 − 3 4 ∑ n = 1 ∞ 1 16 n 2 − 1 1 4 n 2 − 1 {\displaystyle {\begin{aligned}A&={\frac {1}{4}}\cdot {\frac {1}{2}}+\sum _{n=1}^{\infty }\left(1-{\frac {1}{4}}{\frac {\frac {1}{4n^{2}-1}}{\frac {1}{16n^{2}-1}}}\right){\frac {1}{16n^{2}-1}}\\&={\frac {1}{8}}-{\frac {3}{4}}\sum _{n=1}^{\infty }{\frac {1}{16n^{2}-1}}{\frac {1}{4n^{2}-1}}\end{aligned}}}
which converges much faster than the original series. Coming back to Leibniz formula, we obtain a representation of π {\displaystyle \pi } that separates 3 {\displaystyle 3} and involves a fastly converging sum over just the squared even numbers ( 2 n ) 2 {\displaystyle (2n)^{2}} ,
π = 4 − 8 A = 3 + 6 ⋅ ∑ n = 1 ∞ 1 ( 4 ( 2 n ) 2 − 1 ) ( ( 2 n ) 2 − 1 ) = 3 + 2 15 + 2 315 + 6 5005 + ⋯ {\displaystyle {\begin{aligned}\pi &=4-8A\\&=3+6\cdot \sum _{n=1}^{\infty }{\frac {1}{(4(2n)^{2}-1)((2n)^{2}-1)}}\\&=3+{\frac {2}{15}}+{\frac {2}{315}}+{\frac {6}{5005}}+\cdots \end{aligned}}}
See also Euler transform
References
External links Weisstein, Eric W. "Kummer's Series Transformation". MathWorld.