Talk:Cesàro summation

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 Field: Analysis

Sentence doesn't make sense[edit]

The following sentence in the article doesn't make sense: "In fact, any series which diverges to (positive or negative) infinity the Cesàro method also leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable."

Should it say something along the lines of "In fact, any series whose partial sums diverge to plus or minus infinity also leads to a sequence that diverges likewise in the Cesàro sense."?


"...Cesàro summation is an alternative means of assigning a sum...." Ha ha, "means". I hope the pun was intentional. (talk) 23:43, 21 March 2010 (UTC)

difference between Cesàro sum and limit?[edit]

The article currently says that the generalized Cesàro sum can be restated as

(C,\alpha)-\sum_{j=0}^\infty a_j = \lim_{n\to\infty} \sum_{j=0}^n \frac{{n \choose j}}{{n+\alpha \choose j}} a_j.

However, the article also says that if the series converges then the Cesàro sum is equal to the limit, and that the generalized Cesàro sum is just an iterated version of the normal Cesàro sum – this would seem to imply that if \sum_{j=0}^\infty a_j exists then the difference on the left-hand side should be zero. Am I missing something? Joriki (talk) 21:15, 26 November 2010 (UTC)

The LHS is not a difference. It's (bad but standard) notation for the sum. Sławomir Biały (talk) 12:24, 4 February 2011 (UTC)
Currently, the ASCII hyphen is being automatically converted to a minus sign because it's inside <math> tags. Perhaps this would be less confusing if we were to use a non-math hyphen or an en-dash or something? Joule36e5 (talk) 23:54, 1 February 2013 (UTC)
I changed it to an en dash. Joule36e5 (talk) 09:20, 26 February 2015 (UTC)

Irregular oscillator[edit]

"Since a sequence that is ultimately monotonic either converges or diverges to infinity, it follows that a series which is not convergent but Cesàro summable oscillates." Note that it doesn't have to be a regular cycle... For example:
n = {1,2,3,4,5,6,7,8,...} integers greater than 0
x_n = cos(n)+cos(n sqrt(2)) cycle that never repeats because of impossible factoring

This is a trivial example. Since by definition, a transcendental number never repeats, this means that it is impossible to have harmonics with n and n sqrt(2). Since the natural period of the cosine function is a multiple of Pi, some care has to be taken when constructing such examples to not allow the terms inside the cosine functions have certain values (or patterns). n and n*1.41... with n being positive integers satisfies this requirement. (talk) 09:56, 19 January 2012 (UTC)

Sorry, meant to mention my question. Does anyone think this is important enough of a point to add to the article. And I dare someone to prove that the square root of 2 is rational!  ;) (It's an old disproof in any decent book on proofs) (talk) 10:01, 19 January 2012 (UTC)