Talk:Series (mathematics)

Sigma?

Don't use \Sigma instead of \sum . Please note:

$\Sigma _{n=1}^{3}a_{n}\,$

$\sum _{n=1}^{3}a_{n}\,$

The former uses \Sigma ; the latter uses \sum . Michael Hardy 14:39, 11 April 2007 (CDT)

Taylor series

Do we include stuff about Taylor series in this article, or start another one about Taylor series? Yi Zhe Wu 16:41, 21 July 2007 (CDT)

Of course Taylor series deserves an article — and enjoys one, BTW. Surely, it should be briefly announced/linked in the present article, as well as Fourier series. My idea is to go through some convergence criteria as this is the very first task of the analysis and the "motivations" section prepared the ground for this. Next, pass to particularly important cases like power/Taylor series or Fourier one. --Aleksander Stos 17:10, 21 July 2007 (CDT)

Nice, and good job too! I learned about Taylor series this year, but not Fourier series. Best. Yi Zhe Wu 18:50, 21 July 2007 (CDT)

Ratio vs root test

I'm not so sure what the last paragraph wants to say. It may happen that the ratio test does not give an answer, because the limit of ratios does not exist, while the root test shows convergence or divergence. Example: the series 1 + 1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/16 + 1/16 + … Also, the limes superior (lim sup) should perhaps be explained? -- Jitse Niesen 21:14, 22 August 2007 (CDT)

Right you are! I meant that we obtain the same limit L in both cases (when it exists) and the choice between the two criteria is of practical/computational nature, just like you indicate. It was far from being clear. Please fix as you like, otherwise I'll fix it today. Aleksander Stos 02:04, 23 August 2007 (CDT)

I reworked the bad text. On reflection, maybe the \limsup should be simply turned into the \lim? I mean that this is the basic ("most popular") version -- and for ratio test I used a similar approach (there are some limsup/liminf formulations for ratio test as well). I think refinements can be presented in separate articles on tests, here we would adopt basic versions. if no objection appears I'll introduce the simplification. Aleksander Stos 11:32, 23 August 2007 (CDT)

PS Thanks for copy editing.

Clarify what is being summed

It seems to me that the page dances around precisely which numbers are being added. For instance, in the formal definition section, it refers only to "elements that can be added", thereby leaving the possibility of consideration of series of rational numbers, real numbers, complex numbers, p-adic numbers, and more generally, elements of a topological group of some sort. However, in the discussion of alternating series, it says that the elements in a series can change sign, tacitly assuming that the series are now series of real numbers. Clearly, the series that would be found most interesting to a general audience are series of real numbers. However, in writing up the Riemann zeta function page, I pointed out that to understand the definition, one must understand infinite series of complex numbers, so these should be addressed on some page.

It seems to me that addressing this issue would require some sound organizational thought, and then possibly some serious revision of the present article. Any suggestions for the "best" way to approach this issue? Barry R. Smith 21:28, 27 March 2008 (CDT)