If a black hole was near enough us to be seen by telescope (by eye), if it ate a star, how long would we be seeing the star being eaten? Since we are external observers, an object falling in a black hole should take an infinite amount of time right? If so, what is the answer to my question in the title?
 A: While something falling into a black hole doesn't pass the horizon in a finite time, as calculated by an outside observer, the things that have fallen in become unobservable very shortly after they approach the horizon. This is because the redshift from somewhere near the horizon is so large that the photons have an enormous wavelength, and are thus unobservable. The timescale for falling-in matter to be redshifted to invisibility is a small multiple of the time it would take light to travel a distance of $r$, where $r$ is the radius of the black hole. This is on the order of milliseconds for a solar-mass black hole, and hours for the supermassive black holes at the centers of galaxies.
When a real black hole eats a star, how long do we observe this process happening? I expect the number varies greatly depending on the size of the black hole, and the details of how it eats the star. This website says we have actually seen a star being eaten by a black hole. Or at least, the star was ripped apart by the black hole and some of it was eaten. It took around a year. But it took this long not because of the infinite time the star took falling into the black hole once it came close to the event horizon, but because the star was orbiting the black hole, relatively far from the event horizon, while it was being ripped apart. 
A: Well theoretically, the answer to your question would be infinite amount of time, but practically, it would take longer than the actual duration of the event due to slow down of photons. If it was really infinite time, then we should be able to see history of all the falls since a black hole was first formed, which obviously is not true today. Not only that we would be able to see it now, but also we should be able to do so for an infinitely long time in future. Once a photon leaves from near horizon, it would pick up speed pretty fast and should not take that long for the photons to disappear.
Even if there some are photons that take infinitely long time, they would not be sufficient to form the "being Eaten" image for a very long time. I guess we need some minimum number of photons per second to form a continuous image.
Looking at it in another way, If there are certain number of photons crossing the surface of the sphere, for an infinite long time, it would require infinite amount of energy just in terms of light.
