To determine the values of x for which the series converges, we first express the series explicitly. The given series can be interpreted as:
S = ∑ (x * 3^n * 2^n)
This can be rewritten as:
S = x * ∑ (3 * 2)^n
Now, simplify the series term:
S = x * ∑ (6^n)
We recognize that the series ∑ (6^n) is a geometric series with a common ratio of 6. A geometric series converges when the absolute value of the common ratio is less than 1:
|6| < 1
Since this condition is never satisfied for real numbers, the series diverges for all values of x.
Therefore, there are no values of x for which the series converges.