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.