Question

If | x | < 1, then the square root of the sum 1 + 2x + 3x² + 4x³ + .... ∞

(a) (1 – x) 

(b) (1 + x) 

(c) (1 – x) ^{– 1} 

(d) (1 + x) ^{– 1}

Answer 1

(c) (1 – \(x\)) ^–1

Let S∞ = 1 + 2x + 3x² + 4x³ + ..... ∞                    ...(i) 

S_∞ is an infinite A.G.P. with A.P. : 1 + 2 + 3 + .... ∞ 

and G.P. : 1 + \(x\) + x² + x³ + .... ∞                      .......(ii)

The common ratio of the A.G.P. is \(x\) 

∴  \(x\) S_∞ = \(x\) + 2x² + 3x² + .... ∞                   ...(iii) 

Eq (i) – Eq (ii) 

⇒ (1 – \(x\)) S_∞ = 1 + \(x\) + x² + x³ + .... ∞ 

⇒ (1 – \(x\)) S_∞ = \(\frac{1}{1-x}\)                     \(\bigg(\because\,S_\infty=\frac{a}{1-r}\bigg)\)

⇒ S_∞ = \(\frac{1}{(1-x)^2}\) = (1 - x )^-2

∴ Square root of S_∞ = (1 – \(x\)) ^–1