English Dictionary

binomial series

名詞

広義

(mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲𝛼 choose kxᵏ, where 𝛼 choose k=(𝛼(𝛼-1)(𝛼-2)…(𝛼-k+1))/(k!).

日本語の意味

二項級数：数学における用語で、関数 f(x) = (1+x)^α のマクローリン級数展開、すなわち Σₖ₌₀∞ (α choose k)x^k として表される級数を指す。

To approximate fractional powers, the mathematician used the binomial series centered at zero and verified its convergence for |x| < 1.

分数乗を近似するために、数学者は任意の複素数αに対する関数 f(x) = (1 + x)^α のマクローリン級数展開を用い、その収束が |x| < 1 で成り立つことを確認した。

意味（1）

(mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲𝛼 choose kxᵏ, where 𝛼 choose k=(𝛼(𝛼-1)(𝛼-2)…(𝛼-k+1))/(k!).

Thus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1.

意味（2）

(mathematics, loosely) The binomial theorem.

A particular useful series is termed the binomial series (or binomial theorem) and is frequently used to simplify engineering expressions. In this chapter we consider the arithmetic, geometric and binomial series.

binomial series （ plural ）