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A Kronecker's result

Let

Underoverscript[∑, n = 1, arg3] x _ n ,     x _ n ∈ C,(1)

be an infinite series .

L.Kronecker   [1]  proved the following theorem:

Theorem: If (1) is a convergent series and typeset structure, an arbitrary increasing sequence of positive numbers such that typeset structure then

lim _ (n -> ∞) (p _ 1 x _ 1 + p _ 2 x _ 2 + ... + p _ n x _ n)/p _ n = 0.(2)

As it is noted in  [2] , Exercise IV.17.58a, the statement of this theorem is also sufficient:

Theorem: If (2) is true for every increasing sequence of positive numbers  typeset structure,  such that typeset structure, then series (1) is convergent.

An immediate consequence of this result is that the harmonic series is divergent. Take typeset structure, then

FormBox[RowBox[{(p _ 1 x _ 1 + p _ 2 x _ 2 + ... + p _ n x _ n)/p _ n, =, RowBox[{1,    , for,   , every, , RowBox[{n, ., Cell[]}]}]}], TraditionalForm]

References

[1]  Kronecker, L. (1887). Quelques remarques sur la détermination des valeurs moyennes. Comptes Rendus, 103, 980-987.

[2]  Knopp, K. (1947). Theorie und Anwendung der unendlichen Reihen. 4. Aufl. (German). Berlin-Göttingen-Heidelberg : Springer-Verlag. .

Cite this web-page as:

Štefan Porubský: A Kronecker's result.

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