I am struggling with domains, convergence (I think this is Weierstrass Theorem and something else too) and continuity and differentiability.

I am struggling with domains, convergence (I think this is Weierstrass Theorem and something else too) and continuity and differentiability.

For example I am looking for the domain, convergence, and continuity / differentiability of this function.

f(x)= sum from k = 1 to k = infinity, of (x^k + sink) / k^2

Basically I am confused about the sum in the numerator of the fraction - I know separately they converge but I don't understand what that means for this above function. Any help with this would be greatly appreciated. Thank you!