Determine a power series and give the interval of convergence
a). f(x)=x^3/(3-x^2 )
b). f(x)=〖3x〗^2/(5-2∛x)
To determine the power series representation of a function and find its interval of convergence, we can use the technique of polynomial long division and geometric series.
a) Let's find the power series representation of f(x) = x^3 / (3 - x^2).
Step 1: Factorize the denominator.
The denominator 3 - x^2 can be factored as (sqrt(3) + x)(sqrt(3) - x).
Step 2: Rewrite the function.
Rewrite f(x) using the factored denominator:
f(x) = x^3 / ((sqrt(3) + x)(sqrt(3) - x))
Step 3: Perform polynomial long division.
Divide x^3 by (sqrt(3) + x) to get:
x^3 / (sqrt(3) + x) = x(sqrt(3) - x)
Step 4: Rewrite the function again.
Rewrite f(x) as:
f(x) = (x(sqrt(3) - x)) / (sqrt(3) - x)
Step 5: Expand using the geometric series formula.
Using the geometric series formula, we can express (sqrt(3) - x)^-1 as a power series:
(1 / (sqrt(3) - x)) = (1 / sqrt(3)) * (1 / (1 - (-x / sqrt(3))))
The geometric series formula is valid when |x / sqrt(3)| < 1, so the power series representation of f(x) is valid in that interval.
Step 6: Combine the steps.
Combining all the steps, the power series representation of f(x) is:
f(x) = x(sqrt(3) - x) * (1 / sqrt(3)) * ∑ ((-x / sqrt(3)))^n, for |x / sqrt(3)| < 1.
b) Let's find the power series representation of f(x) = (3x^2) / (5 - 2∛x).
Step 1: Rewrite the denominator.
Rewrite the denominator as 5 - 2∛x = 5 - 2x^(1/3).
Step 2: Use polynomial long division.
Divide 3x^2 by (5 - 2x^(1/3)) to get:
(3x^2) / (5 - 2x^(1/3)) = 0.6x^2 + (0.24x^5) / (5 - 2x^(1/3))
Step 3: Expand using the geometric series formula.
We need to express (5 - 2x^(1/3))^(-1) as a power series using the geometric series formula.
However, the geometric series formula is valid only when |x^(1/3) / 2| < 1.
To determine the interval, solve the inequality: |x^(1/3) / 2| < 1.
By squaring both sides, we get: |x^(1/3)| < 4.
Cube both sides to eliminate the cube root: |x| < 64.
Therefore, the interval of convergence for the power series representation of f(x) is: |x| < 64.
Combining all the steps, the power series representation of f(x) is:
f(x) = 0.6x^2 + (0.24x^5) / (5 - 2x^(1/3)), for |x| < 64.