A Note on Derivative of Sine Series with Square Root

Chaundy and Jolliffe proved that if $\left\{a_n\right\}$ is a nonnegative, nonincreasing real sequence, then series $\sum a_n\sin \left(nx\right)$ converges uniformly if and only if $n{a}_n⟶0$ . The purpose of this paper is to show that if $\left\{n{a}_n\right\}$ is nonincreasing and $n{a}_n⟶0$ , then the series $f\left(x\right)=\sum a_n\sin \left(\sqrt{n}x\right)$ can be differentiated term-by-term on $\left[c,d\right]$ for $c,d>0$ . However, $f^{\prime }\left(0\right)$ may not exist.