2.4 (1+x)'/2 = 1 +5* (c) 1-3 3. +. 2.4.6 1-3.5...(2n – 3) +(-1)"+1 2.4.6...(2n) To +.... 1-3 1 x²+ (1+x)-1/2 = 1 - x + 2.4 · · (d) %3D 1.3.5...(2n – 1) + (-1)" - .t. x" +.... 2.4.6...(2n) 9. Prove...

8d:  use the binomial theorem to obtain the following series expansions

1. [Hint: By the binomial theorem, (1+)". 1 < 1+1+="" 1="" 2.3="" 1="" +...+="" 2.3.4...n="" 1="">< 1="" +1+="" 1="" +..="" 22="" 1.="" 2n-1="" 10.="" supply="" the="" missing="" details="" in="" the="" following="" derivation="" of="" newton's="" series="" "/="">
Extracted text: 2.4 (1+x)'/2 = 1 +5* (c) 1-3 3. +. 2.4.6 1-3.5...(2n – 3) +(-1)"+1 2.4.6...(2n) To +.... 1-3 1 x²+ (1+x)-1/2 = 1 - x + 2.4 · · (d) %3D 1.3.5...(2n – 1) + (-1)" - .t. x" +.... 2.4.6...(2n) 9. Prove that the inequality 2 < (1+="" 1/n)"="">< 3="" holds="" for="" all="" integers="" n=""> 1. [Hint: By the binomial theorem, (1+)". 1 < 1+1+="" 1="" 2.3="" 1="" +...+="" 2.3.4...n="" 1="">< 1="" +1+="" 1="" +..="" 22="" 1.="" 2n-1="" 10.="" supply="" the="" missing="" details="" in="" the="" following="" derivation="" of="" newton's="">