Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 3x + 3x^2 + 2x^3 and g(x) = 8 + 16x + 24x^2 + 31x^3 + B. By dividing the power series, find the first few terms of the series for the quotient g(x)/h(x) = Co + C1x + C2x^2 + C3x^3 + … Co = C1 = 2 C2 = 3 C3 = …
The Correct Answer and Explanation is:
Based on the provided power series, the correct coefficients for h(x) are:
c₀ = 4
c₁ = 2
c₂ = 3
c₃ = 4
Explanation
To find the coefficients of the power series for the quotient h(x) = g(x) / f(x), we can use the equivalent relationship g(x) = f(x) * h(x). This allows us to determine the unknown coefficients (c₀, c₁, c₂, c₃, …) by expanding the product of f(x) and h(x) and then equating the coefficients of corresponding powers of x with those in g(x).
The given power series are:
f(x) = 2 + 3x + 3x² + 2x³ + …
g(x) = 8 + 16x + 24x² + 31x³ + …
And the quotient series is:
h(x) = c₀ + c₁x + c₂x² + c₃x³ + …
Setting up the equation g(x) = f(x)h(x):
8 + 16x + 24x² + 31x³ + … = (2 + 3x + 3x² + 2x³ + …)(c₀ + c₁x + c₂x² + c₃x³ + …)
Now, we compare the coefficients for each power of x, solving for one coefficient at a time.
Constant term (x⁰):
The constant term on the left is 8. On the right, it’s the product of the constant terms, 2 * c₀.
8 = 2c₀
c₀ = 4
Coefficient of x (x¹):
The coefficient on the left is 16. On the right, the x term is formed by (2 * c₁x) and (3x * c₀).
16 = 2c₁ + 3c₀
Substitute the value of c₀ = 4:
16 = 2c₁ + 3(4)
16 = 2c₁ + 12
4 = 2c₁
c₁ = 2
Coefficient of x²:
The coefficient on the left is 24. On the right, the x² term is formed by (2 * c₂x²), (3x * c₁x), and (3x² * c₀).
24 = 2c₂ + 3c₁ + 3c₀
Substitute the values of c₀ = 4 and c₁ = 2:
24 = 2c₂ + 3(2) + 3(4)
24 = 2c₂ + 6 + 12
24 = 2c₂ + 18
6 = 2c₂
c₂ = 3
Coefficient of x³:
The coefficient on the left is 31. On the right, the x³ term is formed by (2 * c₃x³), (3x * c₂x²), (3x² * c₁x), and (2x³ * c₀).
31 = 2c₃ + 3c₂ + 3c₁ + 2c₀
Substitute the values of c₀ = 4, c₁ = 2, and c₂ = 3:
31 = 2c₃ + 3(3) + 3(2) + 2(4)
31 = 2c₃ + 9 + 6 + 8
31 = 2c₃ + 23
8 = 2c₃
c₃ = 4
