Answer:
The power series representation for f(x) = 9/(1-x²) centered at x = 0 is: 9 * (1 + x² + x⁴ + x⁶ + ...)
Step-by-step explanation:
To find a power series representation for the function f(x) = 9/(1-x²), we can start by recognizing that the given function is in the form of a geometric series.
A geometric series has the general form of ∑(a * r^n), where a is the first term and r is the common ratio. In this case, the common ratio is x².
We can rewrite the function as:
f(x) = 9 * (1/(1 - x²))
Now, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
Applying this formula to our function, we have:
f(x) = 9 * (1/(1 - x²)) = 9 * (1 + x² + x⁴ + x⁶ + ...)
This gives us a power series representation for the function f(x). The series is centered at x = 0 and the coefficients of the terms are powers of x².
Therefore, the power series representation for f(x) = 9/(1-x²) centered at x = 0 is:
9 * (1 + x² + x⁴ + x⁶ + ...)