Binomial series

A series that arises from expanding (1+x)^n using the binomial theorem.

What is the binomial series?

"In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression..."

What function does the binomial series represent?

"...the binomial series is the Taylor series for the function f(x)=(1+x)^α centered at x=0..."

What is the range of values for α in the binomial series?

"...where α ∈ C (complex numbers)..."

What is the condition for the validity of the binomial series?

"...|x|<1."

How is the binomial series formula expressed?

"The binomial series formula is given by..."

What are the coefficients used in the binomial series called?

"The power series on the right-hand side of (1) is expressed in terms of the (generalized) binomial coefficients."

What is the definition of the binomial coefficients?

"({α k}) := α(α-1)(α-2)...(α-k+1) / k!"

How are the binomial coefficients related to the power series?

"The power series...is expressed in terms of the (generalized) binomial coefficients."

What is the purpose of the Taylor series?

"The binomial series is the Taylor series..."

What does it mean for a nonnegative integer n in the binomial formula?

"For a nonnegative integer n..."

What is the center of the binomial series?

"...centered at x=0."

What is the range for x in the binomial series?

"...|x|<1."

Can α be any complex number?

"Yes, α can be any complex number (α ∈ C)."

How can the binomial series be derived?

"The binomial series is a generalization of the polynomial that comes from a binomial formula expression like...(1+x)^n."

What is the Taylor series expansion of the binomial formula?

"...expressed in terms of the (generalized) binomial coefficients."

What is the mathematical form of the binomial series?

"The binomial series is given by the explicit formula..."

What does the binomial series generalize?

"The binomial series is a generalization of the polynomial..."

What is the polynomial expression associated with the binomial series?

"The binomial series comes from a binomial formula expression like (1+x)^n."

What is the purpose of binomial coefficients in the binomial series?

"The binomial coefficients are used to express the power series in the binomial series formula."

What are the properties of the binomial coefficients?

"The binomial coefficients are defined as a product of factors involving α and k, divided by the factorial of k."