Binomial series

The binomial series is the binomial theorem

On the right side standing power series. Their coefficients are the binomial coefficients, whose name was derived from occurring in the binomial theorem.

Is an integer and, as the number of breaks off to the member and then only consists of a finite sum. For non-integer and for the binomial series delivers the Taylor series of development with point 0

History

The discovery of the binomial series for all positive elements, ie a set formula for numbers of the form can be assigned to Omar Alchaijama from the year 1078 today.

Newton discovered in 1669 that the binomial series for each real number, and all real in the interval represents the binomial. Abel looked at 1826, the binomial series for complex; he proved that it has the radius of convergence 1, if the following holds.

Behavior at the edge of the circle of convergence

It should be and.

• The series converges absolutely if or. [= Real part of α ]
• For all on the edge of the series converges if and only if is.
• For the series converges if and only if or.

Relationship with the geometric series

Substituting and replaced by one obtains

As is, can this number be written as. That is, the binomial series contains the geometric series as a special case.

Examples

• ( a special case of the first binomial formula )

Swell

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

• Sequences and series