>>13463378I know this post is bait but I'll bite anyhow. Infinity always requires a finite treatment in order for humans to argue about it. In the case of power series as commonly defined - yes, this absolutely will cancel for each of the terms.
However, the treatment of infinite sums with real convergence is done with the usual limits, so
Now we can view each of these finite cases with your algebraic identity for all real values as
and so for ,
where I hope it is obvious why only converges when ,
