Answer:
To find the power series representation of the function g(x) centered at c=0, we can expand it as a sum of terms involving powers of (x - 0), which simplifies to x. We'll also simplify the expression:
g(x) = x^2 + 2x - 3 / (4x)
To express g(x) as a power series, we'll focus on the term (2x - 3) / (4x) and expand it separately. We can rewrite it as:
(2x - 3) / (4x) = (1/2) * (x - 3/2x)
Now we'll use the power series representation of (1 - a)^(-1), which is valid for |a| < 1. In our case, a = -3/2x, so we have:
(2x - 3) / (4x) = (1/2) * [1 - (-3/2x)]^(-1)
Expanding this using the power series representation, we get:
(2x - 3) / (4x) = (1/2) * [1 + (-3/2x) + (-3/2x)^2 + (-3/2x)^3 + ...]
Simplifying further:
(2x - 3) / (4x) = (1/2) + (-3/4x) + (9/8x^2) + (-27/16x^3) + ...
Now let's combine this with the x^2 term:
g(x) = x^2 + (1/2) + (-3/4x) + (9/8x^2) + (-27/16x^3) + ...
This is the power series representation of g(x) centered at c=0.
To determine the interval of convergence, we need to find the values of x for which the series converges. In this case, we observe that the power series involves negative powers of x, so it converges when x ≠ 0.
Therefore, the interval of convergence is (-∞, 0) ∪ (0, +∞) (excluding 0), expressed in interval notation.