[obm-l] serie interessante

["Luis Lopes" <llopes@xxxxxxxxxxxxx>]

Sauda,c~oes,

O professor Rousseau acabou de me mandar
o seguinte email:

Dear Luis:

  Thanks.  I am just writing up a solution set for one of
my classes that includes a most remarkable sum.
Unfortunately, I don't know even an Eulerian approach to
this problem; only complex analysis works as far as
I know.  Anyway, here is the series

\sum_{k=1}^{\infty} 1/x_k^2 = 1/10,

where 0 < x_1 < x_2 < x_3 < \cdots are the positive
roots of  the equation  x = \tan x.  What's so remarkable
about this series is that we don't have a precise value for
any one of the terms (although they can be approximated
to many decimal places by numerical computation), but
we do have a precise value (1/10) for the sum.
Something to ponder.

Cecil

Comentário: não sei se os x_k e os 1/x_k^2 são
transcendentes mas se forem, teríamos uma soma
infinita de termos transcendentes dando um número
racional.

[]'s
Luís


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