Question 9.9: Discuss the uniform convergence of the series ∑fn, where (i)...

(i) Suppose D is a subset of \mathbb{R} bounded by K, i.e., |x| ≤ K for all x ∈ D. Then, using Example 7.11,

\left|\sin \left(\frac{x}{n^{2}}\right)\right| ≤ \frac{|x|}{n^{2}} ≤ \frac{K}{n^{2}}  for all x ∈ D.

Taking M_{n} = K/n^{2} and noting that \sum{M_{n}} is convergent, we conclude that \sum{f_{n}} is uniformly (and absolutely) convergent on any bounded subset of \mathbb{R}.

Since, however, f_{n} → 0 pointwise on \mathbb{R} and

\underset{x∈\mathbb{R}}{\sup}|f_{n}(x)| ≥ \left|\sin \left(\frac{n^{2}π/2}{n^{2}}\right)\right| = 1 \nrightarrow 0,

we see that (f_{n}) does not converge uniformly to 0 on \mathbb{R}. Consequently, the series \sum{f_{n}} does not converge uniformly on \mathbb{R}.

(ii) The series \sum{1/n^{2}x^{2}} clearly converges pointwise on the open set \mathbb{R}\setminus \left\{0\right\}. Now let r > 0. For all x ∈ \mathbb{R} such that |x| > r, we have

|f_{n}(x)| ≤ \frac{1}{n^{2}r^{2}}  for all n ∈ \mathbb{N}.

Since \sum{1/n^{2}r^{2}} is convergent, the series \sum{f_{n}} converges uniformly, by the M-test, on the closed set \mathbb{R}\setminus (−r, r) = (−∞, r] ∪ [r, ∞) for all r > 0.

But, though f_{n}(x) → 0 pointwise on \mathbb{R}\setminus \left\{0\right\},

\underset{x≠0}{\sup} |f_{n}(x)| ≥ |f_{n}(1/n)| = 1 \nrightarrow 0.

Hence (f_{n}) does not converge uniformly to 0 on \mathbb{R}\setminus \left\{0\right\}, and \sum{f_{n}} therefore fails to converge uniformly on \mathbb{R}\setminus \left\{0\right\}.

(iii) The series \sum{2^{−n} \sin(3^{n}x)} is easily seen to be uniformly convergent on \mathbb{R} by the M-test, using M_{n} = 2^{−n}. Since f_{n} is continuous on \mathbb{R} for all n, its sum

f (x) = \sum\limits_{n=1}^{∞}{2^{−n}} \sin(3^{n}x)

is also a continuous function on \mathbb{R}, by Theorem 9.7. But, surprisingly, f is nowhere differentiable (see [KRA]). This is an example of a function, first pointed out by Weierstrass, which is continuous everywhere but nowhere differentiable.