We can apply the ratio test to determine the values of x for which the series converges:
lim(n->∞) |(2^(n+1)/x^(n+1)) / (2^n/x^n)| = lim(n->∞) |(2/x)| = 2/|x|
Therefore, the series converges if 2/|x| < 1, or equivalently, if |x| > 2. So the series converges for x > 2 and x < -2.
To find the sum of the series for those values of x, we can use the formula for the sum of a geometric series:
S = a/(1-r)
where a = 2/x and r = 1/x. So we have:
S = (2/x) / (1 - (1/x)) = 2/(x-1)
if x > 2
S = (2/x) / (1 - (1/x)) = 2/(1-x)
if x < -2
Therefore, the sum of the series for x > 2 is 2/(x-1), and the sum of the series for x < -2 is 2/(1-x).