Find the Value of x for which the series Σ n=1 [(3-2x)^n]/(2^n) is convergent

Apply the Ratio Test:

a(n+1)/a(n) = 2^n (3-2x)^(n+1) /2^(n+1) (3-2x)^n
= (3-2x) /2

The series converges if | (3-2x)/2 | < 1
|3-2x| < 2
-2 < (3-2x) < 2

-5 < -2x < -1
5 > 2x > 1
1 < 2x < 5
1/5 < x < 5/2