As the amplitude θ_m in Eq. 17-25 approaches 180°, what value do you expect the period to approach? Explain in physical terms.

The general equation for the time period, when the pendulum oscillating with large amplitude is,

Here is the maximum angular displacement and T increases with increasing amplitude. Succeeding terms in the finite series becomes smaller and smaller. The value of time period is greater than that of the period when the amplitude is small.
When θ_m = 180°, then the pendulum approaches to upper extreme position. So, it may fall or complete the rotation depends upon the speed at upper extreme position. But θ_m = 180° means the speed of pendulum will be zero. Thus it will drop directly down and the movement or oscillation will no longer. Thus oscillation will sink.