 To illustrate this, consider

Then the series formula_4 is uniformly convergent (for any "ε" take "n" ≥ 1/"ε"), but the series of uniform norms is the harmonic series and thus diverges.An example using continuous functions can be made by replacing these functions with bump functions of height 1/"n" and width 1 centered at each natural number "n".As well, normal convergence of a series is different from "norm-topology convergence", i.e.