binomial series
التعريفات والمعاني
== English ==
=== Noun ===
binomial series (plural binomial series)
(mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)α, for arbitrary complex α; the series
∑
k
=
0
∞
(
α
k
)
x
k
{\displaystyle \textstyle \sum _{k=0}^{\infty }{\alpha \choose k}x^{k}}
, where
(
α
k
)
=
α
(
α
−
1
)
(
α
−
2
)
…
(
α
−
k
+
1
)
k
!
{\displaystyle \textstyle {\alpha \choose k}={\frac {\alpha (\alpha -1)(\alpha -2)\dots (\alpha -k+1)}{k!}}}
.
2010, James Stewart, Calculus: Concepts and Contexts, Cengage Learning, page 612,
Thus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1.
(mathematics, loosely) The binomial theorem.
==== Usage notes ====
The infinite series is a direct generalisation of the (finite) binomial theorem expansion of (1 + x)n (n a positive integer): in both cases, the notation
(
α
k
)
{\displaystyle \textstyle {\alpha \choose k}}
, as defined above, is applicable for the coefficients, which are called binomial coefficients. (Note that the binomial theorem treats the slightly different form (x + y)n, which does not directly generalise to an infinite series.)
==== Hypernyms ====
(Maclaurin series expansion of (1 + x)α): Maclaurin series, power series Taylor series
==== Translations ====
=== Anagrams ===
biomineralises