Grandiose Gibberish

Suppose we define a piecewise function, \( \text{frankensine}(x) \) as described below.

\[ \text{frankensine}(x) = \begin{cases} \sin(x) \ \quad, \quad 0 < x < 2\pi \\ \sin(2x) \quad, \quad 2\pi < x < 2\pi\left(1 + \frac12 \right) \\ \sin(3x) \quad, \quad 2\pi\left(1 + \frac12 \right) < x < 2\pi\left(1 + \frac12 + \frac13\right) \\ \sin(4x) \quad, \quad 2\pi\left(1 + \frac12 + \frac13\right) < x < 2\pi\left(1 + \frac12 + \frac13+\frac14\right) \\ \vdots \\ \sin(nx) \quad, \quad 2\pi\left(1 + \frac12 + \frac13+\ldots + \frac1{n-1}\right) < x < 2\pi\left(1 + \frac12 + \frac13+\frac14+\ldots + \frac1n \right) \\ \vdots \end{cases} \]

For integer \(n\), let \[ \displaystyle A = \lim_{n\to\infty} \int_0^n \text{frankensine}(x) \, dx. \]

What is the value of \( \lfloor 1000A \rfloor \)?

Inspiration.