Final answer:
To find a power series centered at the origin that converges to the expression 12/(3+y), we use the formula for the sum of a geometric series. The series converges for all values of y between -9 and 9.
Step-by-step explanation:
To find a power series centered at the origin that converges to the expression 12/(3+y), we will use the formula for the sum of a geometric series. The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
Where S represents the sum of the series, a is the first term, and r is the common ratio. In this case, we can rewrite the expression 12/(3+y) as a power of x. Let's assume that x = -y/3. Then, the expression becomes:
12 / (3 + x)
Now, we can see that the first term of our power series is 12 and the common ratio is -x/3. Plugging these values into the formula, we get:
S = 12 / (1 - (-x/3))
Simplifying further, we get:
S = 12 / (1 + x/3)
This is our power series centered at the origin which converges to the expression 12/(3+y). Now, let's determine the values for which the series converges. The series converges when the absolute value of the common ratio, |-x/3| is less than 1. So, |-x/3| < 1. Solving for x, we find that -1 < x/3 < 1. Multiplying by 3, we get -3 < x < 3. Therefore, the series converges when -3 < y/3 < 3. Multiplying by 3 again, we get -9 < y < 9. So, the series converges for all values of y between -9 and 9.