Step-by-step explanation:
(a) The derivative of xⁿ is n xⁿ⁻¹. Therefore, if ∑ xⁿ = 1 / (1 − x), then:
∑ (n xⁿ⁻¹) = d/dx (1 / (1 − x))
∑ (n xⁿ⁻¹) = 1 / (1 − x)²
(b)(i) ∑ (n xⁿ) = x ∑ (n xⁿ⁻¹) = x / (1 − x)²
(b)(ii) x = 1/9, so the sum is:
1/9 / (1 − 1/9)²
1/9 / (8/9)²
9/64
(c)(i) The derivative of n xⁿ⁻¹ is n (n − 1) xⁿ⁻². Therefore:
∑ (n (n − 1) xⁿ⁻²) = d/dx (1 / (1 − x)²)
∑ (n (n − 1) xⁿ⁻²) = 2 / (1 − x)³
∑ (n (n − 1) xⁿ) = 2x² / (1 − x)³
(c)(ii) x = 1/4, so the sum is:
2 (1/4)² / (1 − 1/4)³
1/8 / (3/4)³
8/27
(c)(iii) ∑ n² / 2ⁿ
∑ (n² − n) / 2ⁿ + ∑ n / 2ⁿ
2 (1/2)² / (1 − 1/2)³ + 1/2 / (1 − 1/2)²
1/2 / 1/8 + 1/2 / 1/4
4 + 2
6
