You're wanting to flash a LED at 1000 Hz, powered by a 9V battery? Why not use a 555 chip? Plenty of circuits to copy, then tweak components to see how they change the result. It will be generating a rectangular waveform, not sinusoidal.
To see the strobing, and to compare ON duration with OFF duration, build everything compactly on one board, then in a darkened room hold the board in your extended arm and swing it vigorously in a wide arc, back and forth as fast as you can, and you'll he able to see the path traced out by the LED comprises lit and unlit portions. 1000 Hz is about the upper limit to see this, so I suggest that you design for around 100 Hz to start with, then once that is operating as desired, just change the R or C timing component(s) to produce your higher frequency.