English Dictionary

Taylor series

名詞

(calculus) A power series representation of given infinitely differentiable function f whose terms are calculated from the function's arbitrary order derivatives at given reference point a; the series f(a)+(f'(a))/(1!)(x-a)+(f(a))/(2!)(x-a)²+(f'(a))/(3!)(x-a)³+⋯=∑ₙ₌₀∞(f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ.

日本語の意味

テイラー級数は、無限回微分可能な関数に対して、その関数の参照点における任意の次数の微分値を用い、関数を多項式の無限級数として表現する方法です。

To approximate sin(x) near zero, students often use the Taylor series to express it as a power series whose coefficients are calculated from the function's derivatives at that point.

sin(x) を 0 付近で近似するために、学生たちはしばしばその点における関数の導関数から係数を求めて表す冪級数、つまりテイラー展開を用いる。

意味（1）

(calculus) A power series representation of given infinitely differentiable function f whose terms are calculated from the function's arbitrary order derivatives at given reference point a; the series f(a)+(f'(a))/(1!)(x-a)+(f(a))/(2!)(x-a)²+(f'(a))/(3!)(x-a)³+⋯=∑ₙ₌₀∞(f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ.

A series solution about an ordinary point of a differential equation is always a Taylor series having a nonvanishing radius of convergence. A series solution about a singular point does not have this form (except in rare cases). Instead, it may be either a convergent series not in Taylor series form (such as a Frobenius series) or it may be a divergent series.

Taylor series （ plural ）