Question

Consider the series f(x)=∑[infinity]ₙ₌₀(n+4)/3ⁿxⁿ

(i) What is the radius of convergence of this series? Write the letter i if the radius is infinite.
(ii) Find the series expansion, centered at x=0, for the derivative f ′(x) of f(x). What is the coefficient of x⁴ in this series? (Do not use the symbol! and give the exact value)
(iii) What is the radius of convergence of the series for f ′(x) ? Write the letter i if the radius is infinite. (iv) Find the series expansion, centered at x=0, for a primitive ∫f(x)dx of f(x). What is the coefficient of x⁴? (Do not use the symbol ! and give the exact value)
(v) What is the radius of convergence of the series for ∫f(x)dx ? Write the letter i if the radius is infinite.

Answer 1

The radius of convergence for the series f(x) is 3. The series expansion for the derivative f'(x) has a coefficient of x⁴ equal to 32/81. The coefficient of x⁴ in the series expansion for the primitive ∫f(x)dx is 7/36. The radius of convergence for both the derivative and the primitive series is also 3.

Step-by-step explanation:

(i) To find the radius of convergence of the series f(x)=∑[infinity]ₙ₌₀(n+4)/3ⁿxⁿ, we can use the ratio test. The ratio test states that if R is the radius of convergence and limₙ₌ₒ|aₙ₊₁/aₙ| = L, then R = 1/L. Taking the ratio of consecutive terms in the series, we have:

limₙ₌ₒ|((n+5)/3) * x/(n+1)| = L.

By simplifying and taking the limit, we find L = |x/3|, so the radius of convergence is R = 3. Therefore, the answer to part (i) is 3.

(ii) To find the series expansion for the derivative f ′(x) of f(x), we can differentiate the given series term by term. Differentiating (n+4)/3ⁿxⁿ, we get:

f ′(x) = ∑[infinity]ₙ₌₀(n+4)/3ⁿ * n * xⁿ⁻¹.

The coefficient of x⁴ in this series is given by the term when n = 4:

a₄ = (4+4)/3⁴ * 4 = 32/81.

Therefore, the coefficient of x⁴ is 32/81.

(iii) The radius of convergence of the series for f ′(x) can be found using the ratio test in the same way as part (i). Taking the ratio of consecutive terms in the series for f ′(x), we have:

limₙ₌ₒ|((n+4)/3ⁿ * n * xⁿ⁻¹)/((n+5)/3ⁿ₊₁ * (n+1) * xⁿ)| = L.

Simplifying and taking the limit, we find L = |x/3|. Therefore, the radius of convergence for f ′(x) is also 3.

(iv) To find the series expansion for the primitive ∫f(x)dx of f(x), we can integrate the given series term by term. Integrating (n+4)/3ⁿxⁿ, we get:

∫f(x)dx = ∑[infinity]ₙ₌₀(n+4)/3ⁿ * xⁿ⁺¹ / (n+1).

The coefficient of x⁴ in this series is given by the term when n = 3:

a₃ = (3+4)/3³ * 1/4 = 7/36.

Therefore, the coefficient of x⁴ is 7/36.

(v) The radius of convergence of the series for ∫f(x)dx can be found using the ratio test in the same way as part (i) and (iii). Taking the ratio of consecutive terms in the series for ∫f(x)dx, we have:

limₙ₌ₒ|((n+4)/3ⁿ * xⁿ⁺¹ / (n+1))/((n+5)/3ⁿ₊₁ * (n+2) * xⁿ₊₁)| = L.

Simplifying and taking the limit, we find L = |x/3|. Therefore, the radius of convergence for ∫f(x)dx is also 3.