Uniform Convergence of Trigonometric Series

Determine if the series $
}

converges uniformly.

In the realm of analysis, understanding the convergence of series, particularly those involving trigonometric functions, is crucial. The problem at hand requires the examination of uniform convergence of the series sum of sine functions divided by an expression involving powers of k. Uniform convergence is a stronger form of convergence than pointwise convergence and has significant implications in analysis, such as the interchangeability of limits and integration or differentiation. To determine uniform convergence of this series, we first delve into the definition of uniform convergence: a series converges uniformly on a set if, for any given positive epsilon, there exists a natural number N such that the absolute difference between the function series and its Nth partial sum is less than epsilon for all indices beyond N, uniformly for all x in the domain.

The series includes a trigonometric sine function whose behavior depends sharply on x while the denominator suggests that for larger values of k, the terms become quite small, pointing towards potential convergence. However, when verifying uniform convergence, it is essential to comprehensively consider the behavior of the terms over the domain of x. In particular, employing the Weierstrass M-Test can be insightful. This test provides a sufficient condition for uniform convergence; if each term of the series is dominated by a corresponding term of a convergent series of constants independent of x, then the original series converges uniformly. By examining an appropriate comparison series that converges and bounding the terms of the sine series, conclusions about its uniform convergence can be drawn. Concepts such as bounding, trigonometric identities, and asymptotic behavior of functions become pivotal in this analysis.