I assume this aresequencesof functions as opposed toseries. In that case, regarding your questions, for pointwise convergence we just need an N dependent on x and n. For uniform convergence we need an N dependent only on n (The epsilon is arbitrary so it makes no sense that it would depend on something). For example in the second one we have the bound from which one sees that one can make the sequence of functions arbitrarily arbtrarily close to 0 independently of any point.