Euler's Mistake

He started with the equation x/(x-1) + x/(1-x) = 0,

rewrote it as 1/( 1-(1/x) ) + x/(1-x) = 0,

found power series for both functions and concluded:

... + 1/x^2 + 1/x + 1 + x + x^2 + ... = 0.

The conclusion is false as taking x = 1 makes it:

... + 1 + 1 + 1 + 1 + 1 + ... = 0.

The mistake is as follows:

1/( 1-(1/x) )'s power series is a geometric series which converges when

abs(1/x) less than 1, or abs(x) is greater than 1.

And, x/(1-x)'s power series is a geometric series which converges when

abs(x) is less than 1.

No value of x let both of them to converge at the same time.