S(x) is the sum of the infinite geometric progression with the first term of 1 and the common ratio of  r = 5*(x-5).


Such a progression converges  if and only if  |r| < 1,  which gives


|5*(x-5)| < 1,   or


|x-5| < ,   or


 < x -5 < ,   or, equivalently


4.8 < x < 5.2.


Answer.  The convergence domain is  the interval  (4.8,5.2).