Question 9.10: Discuss the convergence properties of the two series (i) ∑si...

(i) Since |sin nx| ≤ 1 for all $x ∈ \mathbb{R}$ and $\sum{\frac{1} {n^{2}}}$ is convergent, the series $\sum{\frac{\sin nx}{n^{2}}}$ is uniformly convergent on $\mathbb{R}$ by the M test.

(ii) If  $\left|\frac{\sin nx}{n}\right| ≤ M_{n}$ for all x in some interval I, then $M_{n} ≥ \frac{c}{n},$ where $c = \underset{x∈I}{\sup} |\sin nx| > 0.$ Since $\sum{\frac{1} {n}}$ is divergent, we see that the M-test is not applicable on any interval in $\mathbb{R}.$ We shall use the Dirichlet test to prove uniform convergence on any interval $[a, b] ⊆ (2mπ, (2m + 1)π),  m ∈ \mathbb{Z}.$ The proof relies on the equality

$\sum\limits_{k=1}^{n}{\sin kx} = \frac{\cos \frac{1}{2} x − \cos(n + \frac{1}{2})x}{2 \sin \frac{1}{2} x} ,  x ≠ 2mπ, m ∈ Z,$        (9.16)

which can be proved by induction on n. Setting $u_{k}(x) = \sin kx$ and $U_{n}(x) = \Sigma_{k=1} ^{n}  u_{k}(x),$ we see that

$|U_{n}(x)| ≤ \frac{\left|\cos \frac{1}{2}x\right| + \left|\cos(n + \frac{1}{2})x\right|}{2 \left|\sin \frac{1}{2} x\right|}$

$≤ \frac{1}{\left|\sin \frac{1}{2} x\right|}$

$≤ \max \left\{\frac{1}{\left|\sin \frac{1}{2} a\right|}, \frac{1}{\left|\sin \frac{1}{2}b\right|}\right\}$  for all $n ∈ \mathbb{N},  x ∈ [a, b].$

With $v_{n}(x) = 1/n$ on [a, b], we obtain a decreasing sequence which converges (uniformly) to 0. By Dirichlet’s test the series $\sum{\frac{\sin nx}{n}}$ converges uniformly on [a, b].

Using Fourier expansions (see [CAR], for example), it can be shown that $\sum{\frac{\sin nx}{n}}$ converges pointwise on $\mathbb{R}$ to a function which is periodic in 2π, and which is defined on [−π, π] by

$S(x)= \begin{cases} − \frac{π + x}{2}, & x ∈ [−π, 0) \\ 0, & x= 0 \\ \frac{π − x}{2}, & x ∈ (0, π]. \end{cases}$

Note that $x = 2mπ,  m ∈ \mathbb{Z},$ are points of (jump) discontinuity for the function S. Hence the series $\sum{\frac{\sin nx}{n}}$ of continuous terms cannot converge uniformly in any interval containing such points.