So, sin x is defined as power series with rational coefficients,

The statement "the sum of rational numbers is rational" is a theorem only for finite sums. Stuff gets wonky when you start dealing with infinities and limits - if you're new to analysis, this stuff will defy your intuitions at first.

So rather than being confused by the fact that the infinite sum of rationals can be irrational, you should instead look at a proof of such a fact and think, "Huh, so the infinite sum of rationals CAN be irrational!"

If your line of reasoning held, then every real number would be rational -- think about decimal expansions.

>>8799192
The limit point of a sequence of rational numbers may not be rational

>>8799192
зopич хyитa

>>8799192
Sine doesnt exist xdxdxdxdxd

>>8799192
How would you even construct the Real numbers if this wasn't the case?

>>8799192
>Divided by rational number
>soviets cleverly used power series instead of sliderulers

>>8799192
Applying an infinite number of operations to an infinite number of rational numbers can end up with an irrational number. Taylor series are all infinite series

>>8800556

Dedekind cuts?

>>8800556
Axiomatically, as complete, totally ordered field