Example 8.3.4 (c): Function Series Examples
For the given function we have || f_{n}(x) || = 1/n^{2}. Since 1/n^{2} is a p-series with p = 2, so it converges. Therefore f_{n}(x) converges uniformly and absolutely, by Weierstrass' theorem. But since the individual functions f_{n} are not continuous, the sum is not necessarily continuous. In fact we know that 1/2^{n} = 1 and 1/n^{2} = ^{2}/6 (as we will see later), so f_{n}(x) is either 1 if x < 0, or 0 if x = 0, or ^{2}/6 if x > 0. |