Question

Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f: f(x) = 1 / (1 - x³) = ∑(n=0 to [infinity]) x³ⁿ.

a) ∑(n=0 to [infinity]) 3nx³ⁿ⁻¹
b) ∑(n=0 to [infinity]) nx³ⁿ⁻¹
c) ∑(n=0 to [infinity]) x³ⁿ⁻¹
d) ∑(n=0 to [infinity]) 3n²x³ⁿ⁻¹

Answer 1

To obtain the derivative of the given series expansion, we need to differentiate each term of the series ∑(n=0 to ∞) x³ⁿ using the power rule of differentiation.

Step-by-step explanation:

To obtain the series expansion for the derivative of the function f(x) = 1 / (1 - x³), we need to differentiate the given series expansion term-by-term. The original series expansion is ∑(n=0 to ∞) x³ⁿ.

We can differentiate each term of the series by applying the power rule of differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹. Therefore, differentiating each term of the series, we get ∑(n=0 to ∞) 3n*x³ⁿ⁻¹.

So, the correct option is a) ∑(n=0 to ∞) 3nx³ⁿ⁻¹.