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key term - 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.

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