Did Euler really say, "x^i over all integers i = 0 in all bases."

[limber oar]

Was reading this paper (http://dfns.dyalog.com/n_ratrep.htm) cause it was linked in a stack exchange article, and it said Euler claimed:
Where:

S(x) = ... + x^(3) + x^(2) + x^(1) + x^(0) + x^(-1) + x^(-2) + x^(-3) + ...

Then:

S(x) = 0

I can't find anything about this online, does anyone have a link to any resource about this? Or is this not real?

[snow cedar]

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[proud lion]

is this like, formal language theory

[limber oar]

What do you mean?

[wraith hound]

notation was confusing, but it is also explained int he link you gave

the series in question is

$\sum _{k=-\infty}^\infty x^k$

[trim umbra]

al8628

[wraith hound]

but this alone is ambiguous

the only reasonable way I can make sense of it is thinking of it as a Laurent series

meaning that the series converges if and only if

$\sum _{k=0}^\infty x^k\qquad \mbox{and}\qquad \sum _{k=1}^\infty x^{-k}$

[trim umbra]

al8628

[wraith hound]

converge absolutely

but that won't happen

we would also think of it symmetrically

$\lim _{N\to\infty} \sum _{k=-N}^N x^k$

[trim umbra]

al8628

[wraith hound]

which is like an improper riemann integral

but if x>0 the sum, if finite, must be positive

so the intepretation for such series must be entirely different in euler's consideration

the trick employed here

does not work, because we can do addition with finite quantities