Question

Define the binomial series. What is its radius of convergence?

Short answer

The binomial series is represented as \((1+x)^k = 1 + kx + k(k-1)x^2/2! + k(k-1)(k-2)x^3/3! + ...\) for all x and for any real number k. The radius of convergence of the binomial series is 1.

Step-by-step solution

Define the binomial series

The binomial series is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the ratio. It is represented as follows: \( (1+x)^k = 1 + kx + k(k-1)x^2/2! + k(k-1)(k-2)x^3/3! + ... \) for all x and for any real number k.

Calculate the ratio

To find the radius of convergence, we apply the ratio test, which states that if the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series is less than 1, then the series converges. In the case of the binomial series, we find this limit is equal to \(|x|\), so the radius of convergence is 1.