If you use a re-triggerable monoflop (can be made with a 555 or eg. a 4093) and set the period to say 35..40ms and then let it drive the LED, you will have visual indication when there is a signal to the servo.
Problem is, that there is signal to the servo when it's stopped, unless you drop the signal on purpose.
If you only want it to light when there is movement, you could use a small microcontroller to detect signals that are above or below 1.5ms (but I gather it's a bit over your current skill level).
If it's just for checking whether there's signals going to the servo, the re-triggerable monoflop will do.