A Note on Derivative of Sine Series with Square Root

Chaundy and Jolliffe proved that if a n is a nonnegative, nonincreasing real sequence, then series ∑ a n sin n x converges uniformly if and only if n a n ⟶ 0 . The purpose of this paper is to show that if n a n is nonincreasing and n a n ⟶ 0 , then the series f x = ∑ a n sin n x can be differentiated term-by-term on c , d for c , d > 0 . However, f ′ 0 may not exist.