Question

Consider the function f(x)=4 x⁴(3 x+4)² If Sigma c_{n} xⁿ is a power series representation of f(x), then for n eq 4, c_{n}=.If the interval of convergence

Answer 1

The power series representation of f(x) = 4x⁴(3x+4)² is Σ cₙxⁿ = 36x⁶ + 96x⁵ + 64x⁴.

Step-by-step explanation:

Given the function f(x) = 4x⁴(3x+4)², we want to find the power series representation of f(x). To do this, we expand the function using the binomial theorem. The binomial theorem states that (a + b)ⁿ = an + nan-1b + n(n-1)an-2b² + ...

Applying the binomial theorem to f(x), we have:

• f(x) = 4x⁴(3x+4)²
• = 4x⁴(9x² + 24x + 16)
• = 36x⁶ + 96x⁵ + 64x⁴

Therefore, the power series representation of f(x) is Σ cₙxⁿ = 36x⁶ + 96x⁵ + 64x⁴, where cₙ is the coefficient for each term.