See Taylor series on Wiktionary
{
"etymology_text": "Named after English mathematician Brook Taylor, who formally introduced the series in 1715. The concept was formulated by Scottish mathematician James Gregory.",
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"ref": "1978, [McGraw-Hill], Carl M. Bender, Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, published 1999, page 324:",
"text": "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.",
"type": "quote"
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{
"ref": "1980, Suhas Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis (CRC Press), page 28:",
"text": "The usual procedure for deriving finite-difference equations consists of approximating the derivatives in the differential equation via a truncated Taylor series.",
"type": "quote"
},
{
"ref": "1998, Kenneth L. Judd, Numerical Methods in Economics, The MIT Press, page 197:",
"text": "This function has its only singularity at x = 0, implying that the radius of convergence for the Taylor series around x = 1 is only unity.",
"type": "quote"
}
],
"glosses": [
"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."
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"sense": "power series of a function calculated from derivatives at a reference point",
"word": "Maclaurin series"
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"id": "en-Taylor_series-en-noun-kmWUJUxq",
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"(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."
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"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "power series of a function calculated from derivatives at a reference point",
"word": "泰勒展開式"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "Tàilè zhǎnkāishì",
"sense": "power series of a function calculated from derivatives at a reference point",
"word": "泰勒展开式"
},
{
"code": "cs",
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"sense": "power series of a function calculated from derivatives at a reference point",
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"feminine"
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"word": "Taylorova řada"
},
{
"code": "da",
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"sense": "power series of a function calculated from derivatives at a reference point",
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{
"code": "de",
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"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
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"word": "Taylorreihe"
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"sense": "power series of a function calculated from derivatives at a reference point",
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"lang": "Italian",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
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"word": "serie di Taylor"
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"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
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"word": "serie Taylor"
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"code": "ru",
"lang": "Russian",
"roman": "rjad Tɛ́jlora",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"masculine"
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"word": "ряд Те́йлора"
}
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"word": "Taylor series"
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"text": "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.",
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{
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"text": "The usual procedure for deriving finite-difference equations consists of approximating the derivatives in the differential equation via a truncated Taylor series.",
"type": "quote"
},
{
"ref": "1998, Kenneth L. Judd, Numerical Methods in Economics, The MIT Press, page 197:",
"text": "This function has its only singularity at x = 0, implying that the radius of convergence for the Taylor series around x = 1 is only unity.",
"type": "quote"
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],
"glosses": [
"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."
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"(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."
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"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "power series of a function calculated from derivatives at a reference point",
"word": "泰勒展開式"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "Tàilè zhǎnkāishì",
"sense": "power series of a function calculated from derivatives at a reference point",
"word": "泰勒展开式"
},
{
"code": "cs",
"lang": "Czech",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"feminine"
],
"word": "Taylorova řada"
},
{
"code": "da",
"lang": "Danish",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"common-gender"
],
"word": "Taylorrække"
},
{
"code": "de",
"lang": "German",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"feminine"
],
"word": "Taylorreihe"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "power series of a function calculated from derivatives at a reference point",
"word": "Taylor-sor"
},
{
"code": "it",
"lang": "Italian",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"feminine"
],
"word": "serie di Taylor"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"feminine"
],
"word": "serie Taylor"
},
{
"code": "ru",
"lang": "Russian",
"roman": "rjad Tɛ́jlora",
"sense": "power series of a function calculated from derivatives at a reference point",
"tags": [
"masculine"
],
"word": "ряд Те́йлора"
}
],
"word": "Taylor series"
}
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