I encountered this problem while studying for an analysis exam. Here is a related question I asked some days ago.
The problem is as follows: Suppose $a_n$ is a decreasing sequence of positive real numbers and that$\sum_{n = 0}^{\infty}{a_n \sin{(nx)}}$ converges uniformly on $\mathbb{R}$, show that $\lim_{n \to \infty}{(n a_n)} = 0.$ Any tip or solution is welcome, and also avoid using Fourier series, because they haven't been introduced in the book so it can be solved without using them.
Uniform convergence problem
7
$\begingroup$
real-analysis
sequences-and-series
convergence-divergence
-
0@Peláez: No harm done. (English is not my native language either...) – 2011-03-26
1 Answers
10
$\sum_{i=[(k+1)/2]}^k a_i \sin(ix)$ goes uniformly to 0 as $k\to\infty$. Set $x=\pi/(2k)$. Then all $a_i$'s are $\geq a_k$, all the sines are $\geq1/\sqrt{2}$, hence the sum is $\geq (k-1)/2\times a_k/\sqrt{2}$. Since this goes to 0, so does $k a_k$
-
0@user8268 could you explain your answer please? Explain the motivation for letting $i = \frac{k+1}{2}$, how $a_i \geq a_k$, etc. Thanks – 2015-02-25