Answer:
∑ₙ₌₀°° 8 (-1)ⁿ x⁴ⁿ⁺¹ / (9ⁿ (2n)!)
Step-by-step explanation:
Start with the Maclaurin series for cos x.
cos x = ∑ₙ₌₀°° (-1)ⁿ x²ⁿ / (2n)!
Substitute ⅓ x².
cos (⅓ x²) = ∑ₙ₌₀°° (-1)ⁿ (⅓ x²)²ⁿ / (2n)!
cos (⅓ x²) = ∑ₙ₌₀°° (-1)ⁿ (⅓)²ⁿ x⁴ⁿ / (2n)!
cos (⅓ x²) = ∑ₙ₌₀°° (-1)ⁿ x⁴ⁿ / (9ⁿ (2n)!)
Multiply by 8x.
8x cos (⅓ x²) = ∑ₙ₌₀°° 8x (-1)ⁿ x⁴ⁿ / (9ⁿ (2n)!)
8x cos (⅓ x²) = ∑ₙ₌₀°° 8 (-1)ⁿ x⁴ⁿ⁺¹ / (9ⁿ (2n)!)
