Is sequence $x_n=\sum_{i=1}^n \sin(i)$ a Cauchy Sequance?

Question

Is $x_n=\sin(1)+\sin(2)+...+\sin(n)=\sum_{i=1}^n \sin(i)$ a Cauchy sequence?

By defenition $|x_{p+m}-x_p|=|sin(p+1)+...+sin(p+m)|=|cos(p+1/2)-cos(p+m+1/2)|/sin(1/2)$

Answer 1

If $x_n$ is Cauchy then there exists $\epsilon > 0$ and $N$ such that for $m,n > N$ then $|x_n - x_m| < \epsilon$.

But w.l.g. if $n > m$ then $|x_n - x_m| = \sin (m+1) + ... + \sin (n) = \sum_{i=m+1}^n \sin (i) $ which diverges as $n >> m$.