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 .
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.
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.
Behavior at the edge of the convergence circle
It be and .
The series converges absolutely if and only if or is. [ = Real part of α]
For all on the edge the series converges if and only if is.
For the series converges if and only if or is.
Relationship to the geometric series
If you place and replace with , you get
Since is, this series can also be written as . That is, the binomial series contains the geometric series as a special case.