Time Dilation and Particle Decay I am teaching myself astronomy and I'm working on a basic problem regarding time dilation around black holes.

The question is basically this: if a neutron were ejected from a nucleus at a distance of 3km from a 1 solar mass black hole, how long would it appear to take for the neutron to decay (the assumption here is that the neutron would decay in 15 minutes... right on the half-life) to an outside observer?

My first thought was that we'd never see it decay, because 3km is the Schwarzschild radius of a BH of that mass, so the frequency would basically be $0$ and we'd never observe the decay event.

The text notes the time dilation 

$$
\frac{\Delta t'}{\Delta t_\text{obs}} = \frac{\nu_f}{\nu_i}$$
where $\Delta t'$ is the the time difference between two events at the source and $\Delta t_{obs}$ at the observer.

We can use the relativistic red shift equation to get the wavelength difference (9.2 in my case using 1 solar mass and a radius of 3km) so that wouldn't be infinite, just big. For a 15 min $\Delta t'$ then, the $\Delta t_{obs}$ would be 15 min * 9.2, yeah?

So my two answers obviously contradict each other. What am I missing in my approach here?
If you use the equation of the Schwarzschild metric.
$$ds^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}$$ 
(equation 11.1 from Schutz's book A first course in GR).

If we think the neutron stay at rest in $r=r_{S}=2M$ then between the born and dead of the neutron $dr=0$ and $d\Omega=0$, and $ds$ will be the proper time, which I will call $\tau$ (I'm folowing the convention $c=1=G$).

Putting this on the metric we have

$$  d\tau^{2}=-(1-\frac{2M}{r})dt_{obs}^{2}$$

$$  \frac{d\tau^{2}}{dt_{obs}^{2}}=-(1-\frac{2M}{r})$$

and if we put $r=r_{s}$ (where the neutros was), we have

$$  \frac{d\tau^{2}}{dt_{obs}^{2}}=0$$

Then $\Delta t_{obs}=\infty$ . 

I couldn't write a comment that's  why I answered, but I think you had some mistake using the red shift equation to get the wavelength difference.

Can you show your calculus? May be I can help.