To determine for which values of x the series Σ (x * (3^n) / (2^n)) converges, we can rewrite the expression inside the summation.
The series can be expressed as:
Σ (x * (3^n) / (2^n)) = x * Σ (3/2)^n
Now, we analyze the series Σ (3/2)^n.
This is a geometric series with a common ratio of r = 3/2. A geometric series converges if the absolute value of the common ratio is less than 1:
|r| < 1
In our case:
|3/2| = 1.5 > 1
This tells us that the series Σ (3/2)^n diverges.
However, if we want the entire series Σ (x * (3/2)^n) to converge, we must explore the behavior of the series as a whole. Since the geometric series diverges, it means that:
x must be equal to 0 for the entire series to converge. If x = 0, the terms of the series become:
Σ 0 = 0, which converges.
Thus, the only value of x for which the series converges is:
x = 0
In conclusion, the series x * (3^n) / (2^n) converges only when x = 0.