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$1=\sum_{n>0} (nx)^n$ Dont have any solution in $\mathbb C$? Are there other types of equations with no solutions in $\mathbb C$? Can one define an object that satisfyes these equations? Does it behave well?

If x is a matrix, and 1 the identity matrix, is there a solution?

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    Since the RHS series diverges for nonzero $x$, it can hardly be said to be a function of $x$ in $\mathbb{C}$.2012-04-29
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    x isnt restricted to C, so it doesnt have to diverge.2012-04-29
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    You asked about "any solution in $\mathbb{C}$", so I naturally thought you posed the question on that domain. What domain are you asking about?2012-04-29
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    I am asking what domains contain an x which satisfy it.2012-04-29
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    For the question to be meaningful one needs a context in which the "equation" makes sense, presumably at least a topological ring with $\mathbb{N}$ embedded in it (to make sense of the coefficient $n$ equally as the "exponents" of the series).2012-04-29

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