Answer
$ a.\quad$After a few (4) approximations, 1.5707963267949 repeats to the 13th decimal place.
$ b.\quad$
The line segments $(\cos x_{n})$, when very small, almost equal in length the arc length needed in the sum with $x_{n}$ to yield a quarter-circle arc of length $\pi/2.$
(See image.)
Work Step by Step
$ a.\quad$
Set up a spreadsheet, placing 1 in cell A1.
In cell A2, enter the formula =A1+COS(A1) and copy-paste down column A.
This results in:
1
1.54030230586814
1.57079160102426
1.5707963267949
1.5707963267949
1.5707963267949
(Entering "=pi()/2" into a cell, produces 1.5707963267949.)
$ b.\quad$
With each next approximation (brown segments), the $\cos x_{n}$ (horizontal line segment) gets very close to the length of the (red) arc needed to complete (blue) $x_{n}$ arc to $\pi/2$, the quarter-circumference of the unit circle. See image.
