Binomial series

The binomial series is a power series that appears in the binomial theorem for complex numbers . it is

${\displaystyle (x+y)^{\alpha }=\sum _{k=0}^{\infty }{\alpha \choose k}x^{\alpha -k}y^{k}}$.

If and is an integer , the series breaks off after the term and then consists only of a finite sum. Their coefficients are the binomial coefficients , the name of which was derived from their occurrence in the binomial theorem. They calculate to each other , where the falling denotes the factorial . ${\displaystyle \alpha }$ ${\displaystyle \alpha \geq 0}$ ${\displaystyle k=\alpha }$ ${\displaystyle {\alpha \choose k}={\frac {a^{\underline {k}}}{k!}}}$ ${\displaystyle a^{\underline {k}}}$

A special case of the binomial series is the Maclaurin series of the function with ${\displaystyle f}$ ${\displaystyle f(x)=(1+x)^{\alpha },\;|x|<1,\,\alpha \in \mathbb {C} }$

${\displaystyle (1+x)^{\alpha }=\sum _{k=0}^{\infty }{\alpha \choose k}\cdot x^{k}}$.

history

The discovery of the binomial series for whole positive elements, i.e. H. a series formula for numbers of the form can be assigned to Omar Alchaijam from the year 1078 today. ${\displaystyle (a+b)^{n}}$

Newton discovered in 1669 that the binomial series for each real number and all real in the interval , the binomial represents. Abel looked at the complex binomial series in 1826 ; he proved that it has the radius of convergence 1 if holds. ${\displaystyle \alpha }$ ${\displaystyle x}$ ${\displaystyle \left]-1,1\right[}$ ${\displaystyle (1+x)^{\alpha }}$ ${\displaystyle \alpha ,x\in \mathbb {C} }$ ${\displaystyle \alpha \in \mathbb {C} \setminus \mathbb {N} }$

Behavior at the edge of the convergence circle

It be and . ${\displaystyle |x|=1}$ ${\displaystyle \alpha \in \mathbb {C} \setminus \mathbb {N} }$

• The series converges absolutely if and only if or is. [ = Real part of α] ${\displaystyle \textstyle \sum _{k=0}^{\infty }{\alpha \choose k}x^{k}}$ ${\displaystyle \mathrm {Re} (\alpha )>0}$ ${\displaystyle \alpha =0}$ ${\displaystyle \mathrm {Re} (\alpha )}$
• For all on the edge the series converges if and only if is. ${\displaystyle x\neq -1}$ ${\displaystyle \mathrm {Re} (\alpha )>-1}$
• For the series converges if and only if or is. ${\displaystyle x=-1}$ ${\displaystyle \mathrm {Re} (\alpha )>0}$ ${\displaystyle \alpha =0}$

Relationship to the geometric series

If you place and replace with , you get ${\displaystyle \alpha =-1}$ ${\displaystyle x}$ ${\displaystyle -x}$

${\displaystyle {\frac {1}{1-x}}=\sum _{k=0}^{\infty }{-1 \choose k}(-x)^{k}\,.}$

Since is, this series can also be written as . That is, the binomial series contains the geometric series as a special case. ${\displaystyle {\tbinom {-1}{k}}=(-1)^{k}}$ ${\displaystyle \textstyle \sum _{k=0}^{\infty }x^{k}}$

Examples

• ${\displaystyle (1+x)^{2}={\binom {2}{0}}x^{0}+{\binom {2}{1}}x^{1}+{\binom {2}{2}}x^{2}=1+2x+x^{2}}$
  (a special case of the first binomial formula )
• ${\displaystyle {\frac {1}{1+x}}=(1+x)^{-1}=\sum _{k=0}^{\infty }{\binom {-1}{k}}x^{k}=\sum _{k=0}^{\infty }(-x)^{k}}$
• ${\displaystyle {\sqrt {1+x}}=(1+x)^{1/2}=\sum _{k=0}^{\infty }{\binom {1/2}{k}}x^{k}}$

swell

• Otto Forster : Analysis Volume 1: Differential and integral calculus of a variable. Vieweg-Verlag, 8th edition 2006, ISBN 3-528-67224-2 .

Individual evidence

1. ↑ Eric W. Weissenstein: Binomial Series. In: MathWorld - A Wolfram Web Resource. Accessed July 10, 2019 . 
2. ↑ ibid.