Binomial series

Definition

The binomial series is the Taylor series expansion of the function $(1 + x)^n$ around $x = 0$. It generalizes the binomial theorem to cases where the exponent $n$ is not necessarily an integer.

5 Must Know Facts For Your Next Test

1. The general form of the binomial series for $(1 + x)^n$ is $\sum_{k=0}^{\infty} \binom{n}{k} x^k$, where $\binom{n}{k}$ is the generalized binomial coefficient.
2. For integer $n$, the binomial series terminates after a finite number of terms.
3. The radius of convergence for the binomial series $ (1 + x)^n $ depends on whether $ n $ is an integer or not: if $ n \in \mathbb{Z}$, it converges for all $ x $, otherwise it converges for $ |x| < 1 $.
4. The generalized binomial coefficient $\binom{n}{k}$ can be computed using $\frac{n(n-1)(n-2)...(n-k+1)}{k!}$ for any real or complex number $ n $.
5. In applications, the binomial series is useful in approximating functions near a specific point and solving differential equations.

Review Questions

• Write down the first four terms of the binomial series expansion for $(1 + x)^{3/2}$.
• What are the conditions under which the binomial series converges?
• How do you compute a generalized binomial coefficient $\binom{n}{k}$?

Related terms

Taylor Series:

A representation of a function as an infinite sum of terms calculated from its derivatives at a single point.

Power Series:

An infinite series of the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where each term involves powers of $(x - c)$ and coefficients $a_n$.

Binomial Theorem: A formula that provides the expanded form of $(a + b)^n$ as a sum involving terms containing coefficients given by binomial coefficients.