Matematické Fórum

Tato řada diverguje, t.j. nedá se sečíst.
Nutnou (i když nikoliv postačující) podmínkou k tomu, aby řada   konvergovala (tj. dala se jednoznačným způsobem sečíst
a součet měl konečnou hodnotu)  je .  Vidíme, že u Tvé řady tato podmínka splněna není.

Aboj ↑ Myska:,
Mozes nam napisat aky je vseobeobecny clen tvojej rady?

Ahoj ↑ Myska:,
Poznas Cesaro-vu metodu?

Ahoj ↑ Myska:,
Tu je nieco zaujimave
http://en.wikipedia.org/wiki/1_−_ … _%2B_·_·_·
tato cast:
[edit]Cesàro and Hölder


Data about the (H, 2) sum of 1⁄4
To find the (C, 1) Cesàro sum of 1 − 2 + 3 − 4 + ..., if it exists, one needs to compute the arithmetic means of the partial sums of the series. The partial sums are:
1, −1, 2, −2, 3, −3, ...,
and the arithmetic means of these partial sums are:
1, 0, 2⁄3, 0, 3⁄5, 0, 4⁄7, ....
This sequence of means does not converge, so 1 − 2 + 3 − 4 + ... is not Cesàro summable.
There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of (H, n) methods for natural numbers n. The (H, 1) sum is Cesàro summation, and higher methods repeat the computation of means. Above, the even means converge to 1⁄2, while the odd means are all equal to 0, so the means of the means converge to the average of 0 and 1⁄2, namely 1⁄4.[6] So 1 − 2 + 3 − 4 + ... is (H, 2) summable to 1⁄4.
The "H" stands for Otto Hölder, who first proved in 1882 what mathematicians now think of as the connection between Abel summation and (H, n) summation; 1 − 2 + 3 − 4 + ... was his first example.[7] The fact that 1⁄4 is the (H, 2) sum of 1 − 2 + 3 − 4 + ... guarantees that it is the Abel sum as well; this will also be proved directly below.
The other commonly formulated generalization of Cesàro summation is the sequence of (C, n) methods. It has been proven that (C, n) summation and (H, n) summation always give the same results, but they have different historical backgrounds. In 1887, Cesàro came close to stating the definition of (C, n) summation, but he gave only a few examples. In particular, he summed 1 − 2 + 3 − 4 + ..., to 1⁄4 by a method that may be rephrased as (C, n) but was not justified as such at the time. He formally defined the (C, n) methods in 1890 in order to state his theorem that the Cauchy product of a (C, n)-summable series and a (C, m)-summable series is (C, m + n + 1)-summable.[8]

Zda sa mi ze aj Hardy to studuje v jeho knihe Divergent series.
Ak sa mi podari k nej dostat napisem tu ak je tam este nieco zaujimave, co sa tyka toto problemu.

↑ Rumburak:
Teď už mi to je mnohem jasnější, díky :-))

↑ vanok:

Díky za tip, Hardy je klasika.  Kdysi jsem cosi studoval z knihy o řadách od autorů Hardy, Littlewood,
ale bohužel už vůbec nevím, o co tehdy šlo .  )-:-)

↑ vanok:

Ne ne, tato kniha to určitě nebyla, šlo o monografii o řadách (možná speciálně o řadách trigometrických, ale fakt nevím). 
Naštětí tuším, kde v případě potřeby objevit její stopu ...