Editing 2605: Taylor Series
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In mathematics, a {{w|Taylor series}} {{w|Polynomial expansion|expansion}} is a {{w|polynomial}} {{w|power series}} approximation of a function[https://matheducators.stackexchange.com/a/10212] around a given point, composed of an infinite sum of the function's {{w|Derivative|derivatives}}, each both divided by successive {{w|Factorial|factorials}} and multiplied by the incrementally increasing {{w|Exponentiation|power}} of the distance from the given point. Such expansions usually continue without end. Beyond approximation of functions, Taylor series are also useful for deriving numerical approximations of {{w|Irrational number|irrational}} values, {{w|Machin-like formula|such as π}}, as well as {{w|Symbolic integration|symbolic}} forms to make functions easier to integrate or otherwise manipulate with calculus.[https://www.mathsisfun.com/algebra/taylor-series.html] However, because they involve difficult calculus operations, and can be annoyingly tedious to {{w|Numerical analysis|calculate by hand}}, they are often not loved by math students.[https://www.reddit.com/r/EngineeringStudents/comments/gbo8tm/taylor_series_can_fuck_off/] | In mathematics, a {{w|Taylor series}} {{w|Polynomial expansion|expansion}} is a {{w|polynomial}} {{w|power series}} approximation of a function[https://matheducators.stackexchange.com/a/10212] around a given point, composed of an infinite sum of the function's {{w|Derivative|derivatives}}, each both divided by successive {{w|Factorial|factorials}} and multiplied by the incrementally increasing {{w|Exponentiation|power}} of the distance from the given point. Such expansions usually continue without end. Beyond approximation of functions, Taylor series are also useful for deriving numerical approximations of {{w|Irrational number|irrational}} values, {{w|Machin-like formula|such as π}}, as well as {{w|Symbolic integration|symbolic}} forms to make functions easier to integrate or otherwise manipulate with calculus.[https://www.mathsisfun.com/algebra/taylor-series.html] However, because they involve difficult calculus operations, and can be annoyingly tedious to {{w|Numerical analysis|calculate by hand}}, they are often not loved by math students.[https://www.reddit.com/r/EngineeringStudents/comments/gbo8tm/taylor_series_can_fuck_off/] | ||
| − | [[Miss Lenhart]] appears to be teaching a class about how to use a Taylor series. She presumes her students want to keep learning about the series, in that they, "wish it would never end." She then says "Good news!" because the series does not end. The cartoon's humor is based on the contrast between wishing the series won't end, ordinarily desired of sequences of enjoyable events, | + | [[Miss Lenhart]] appears to be teaching a class about how to use a Taylor series. She presumes her students want to keep learning about the series, in that they, "wish it would never end." She then says "Good news!" because the series does not end. The cartoon's humor is based on the contrast between wishing the series won't end, ordinarily desired of sequences of enjoyable events, with the infinite nature of the Taylor series, which is less likely appreciated by her students struggling to understand why the sums {{w|Convergent series|converge}} to their resulting value. |
The title text is a reference to the common practice among physicists and engineers of abbreviating the Taylor series to only the first few terms, typically one or two, in order to simplify the mathematics of their models. The title text is also a pun on the use of the word "series" to refer to a television program. It symbolizes the terms of the mathematical series as a {{w|metaphor}} with a television season, suggesting that only the first term is useful. It makes fun of the common sentiment against bad {{w|screenwriting}} of a series by saying that, "The series should have been cancelled after the first season," replacing "season" with "term." (Notably, both "term" and "season" are used to refer to a stretch of time during which a program is airing—generally, a scholastic or television program, respectively.) Also note that US President {{w|Zachary Taylor}} died during his first term. In a way, his presidency was cancelled during his first term. | The title text is a reference to the common practice among physicists and engineers of abbreviating the Taylor series to only the first few terms, typically one or two, in order to simplify the mathematics of their models. The title text is also a pun on the use of the word "series" to refer to a television program. It symbolizes the terms of the mathematical series as a {{w|metaphor}} with a television season, suggesting that only the first term is useful. It makes fun of the common sentiment against bad {{w|screenwriting}} of a series by saying that, "The series should have been cancelled after the first season," replacing "season" with "term." (Notably, both "term" and "season" are used to refer to a stretch of time during which a program is airing—generally, a scholastic or television program, respectively.) Also note that US President {{w|Zachary Taylor}} died during his first term. In a way, his presidency was cancelled during his first term. | ||
