Talk:Taylor series
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For the "List of Taylor series" I would like to have the first few terms of each series written out for quick reference. I could doit myself, but I don't want to mess anything up.
"Note that there are examples of infinitely often differentiable functions f(x) whose Taylor series converge but are not equal to f(x). For instance, all the derivatives of f(x) = exp(-1/x²) are zero at x = 0, so the Taylor series of f(x) is zero, and its radius of convergence is infinite, even though the function most definitely is not zero."
f(x) has no Taylor series for a=0, since f(0) is not defined. You have to state explicitly that you've defined f(x)=exp(-1/x²) for x not equal to 0 and f(0)=0 . This is merely lim[x->0] f(x), but it is a requirement for rigor.
- Don't complain, fix! Wikipedia:Be bold in editing pages. -- Tim Starling 02:03 16 Jun 2003 (UTC)
By the way, would people call <math> \sum_{n=0}^{\infin} \sum_{p=0}^{\infin} \frac{\partial^n}{\partial x^n} \frac{\partial^p}{\partial y^p} \frac{f(a,b)}{n!p!} (x-a)^n(y-b)^p <math> a Taylor series? Or does it have a name at all? If someone said something about a Taylor series of a 2D (or n D) function, I'd guess they meant something like that... Also, can the term analytic function refer to a 2D function? Κσυπ Cyp 19:01, 17 Oct 2003 (UTC)
- 1st question: sure, see e.g. http://www.csit.fsu.edu/~erlebach/course/numPDE1_f2001/norms.pdf - Patrick 19:51, 17 Oct 2003 (UTC)
- I just shoved it quickly into the article, at the bottom. Κσυπ Cyp 21:49, 17 Oct 2003 (UTC)
Shouldn't the article include something about the "Taylor" for whom the series are named? If I knew, I'd do it myself Dukeofomnium 16:41, 5 Mar 2004 (UTC)
- Good idea. Often a good way to start investigating such things is to click the "What links here" link on the article page. In this case, that reveals that the Brook Taylor page links to the article. -- Dominus 19:00, 5 Mar 2004 (UTC)
What is a "formulat"? it's on the last line. A typo or a word I'm unfamiliar with? Goodralph 16:28, 2 Apr 2004 (UTC)
Edited the geometric series to include cases where n might not start from zero. Stealth 17:22, Feb 19, 2005 (UTC)
In the Taylor series formula, what happens if x=a? when n=0 we get 0 raised to the 0th power, which is undefined. The formula is correct if we define 0^0=1.