Mention the condition of validity of the expansion 8 + x 4 3 and expand up to 4 terms.

Given, 8 + x 4 3 = 8 4 3 1 + x 8 4 3 = 16 1 + x 8 4 3 ∵ 8 4 3 = 8 1 3 4 = 2 4 Expansion of ( 1 + x ) n , where n is fraction exists if and only if | x | < 1 . ∴ 8 + x 4 3 = 16 1 + x 8 4 3 exists if and only if x 8 < 1 i.e., | x | < 8 ∵ | 8 | = 8 Binomial theorem for fractional index, 1 + x n = 1 + n x + n ( n - 1 ) 2 ! x 2 + n ( n - 1 ) ( n - 2 ) 3 ! x 3 + ⋯ ∞ Here, x < 1 and n is a fraction. ∴ 1 + x 8 4 3 = 1 + 4 3 x 8 + 4 3 4 3 - 1 2 ! x 8 2 + 4 3 4 3 - 1 4 3 - 2 3 ! x 8 3 + ⋯ · ∞ = 1 + x 6 + 4 3 · 1 3 2 x 2 64 + 4 3 · 1 3 · - 2 3 6 x 3 512 + ⋯ ∞ = 1 + x 6 + 4 x 2 9 · 2 · 64 - 8 x 3 27 · 6 · 512 + ⋯ ∞ ∴ 8 + x 4 3 = 16 1 + x 6 + x 2 288 - x 3 10368 + ⋯ ∞

Mention the condition of validity of the expansion 9 + 2 x and expand up to 4 terms.

Given, 9 + 2 x = 9 + 2 x 1 2 = 9 1 2 1 + 2 x 9 1 2 = 3 1 + 2 x 9 1 2 Expansion of 1 + x n , where n is a fraction exists if and only if | x | < 1 ∴ 9 + 2 x = 3 1 + 2 x 9 1 2 exists if and only if 2 x 9 < 1 i.e., | x | < 9 2 Binomial theorem for fractional index, 1 + x n = 1 + n x + n ( n - 1 ) 2 ! x 2 + n ( n - 1 ) ( n - 2 ) 3 ! x 3 + ⋯ ∞ Here, | x | < 1 and n is a fraction. ∴ 1 + 2 x 9 1 2 = 1 + 1 2 2 x 9 + 1 2 1 2 - 1 2 ! 2 x 9 2 + 1 2 1 2 - 1 1 2 - 2 3 ! 2 x 9 3 + ⋯ ∞ = 1 + x 9 - 1 2 · 1 2 2 · 4 x 2 36 + 1 2 · 1 2 · 3 2 6 · 8 x 3 729 + ⋯ ∞ = 1 + x 9 - x 2 72 + x 3 1458 + ⋯ · ∞ ∴ 9 + 2 x = 3 1 + x 9 - x 2 72 + x 5 1458 + ⋯ ∞

Mention the condition of validity of the expansion 9 - 4 x 2 and expand up to 4 terms.

Given, 9 - 4 x 2 = 9 - 4 x 2 1 2 = 9 1 2 1 - 4 x 2 9 1 2 = 3 1 - 4 x 2 9 1 2 Expansion of ( 1 + x ) n , where n is fraction exists if and only if | x | < 1 ∴ 9 - 4 x 2 = 3 1 - 4 x 2 9 1 2 exists if and only if 4 x 2 9 < 1 i.e., x 2 < 9 4 ⇒ x 2 < 9 4 [ ∵ x is always > 0 ] ⇒ x < 3 2 Binomial theorem for fractional index, 1 + x n = 1 + n x + n ( n - 1 ) 2 ! x 2 + n ( n - 1 ) ( n - 2 ) 3 ! x 3 + ⋯ ∞ Here, | x | < 1 and n is a fraction. ∴ 1 - 4 x 2 9 1 2 = 1 + 1 2 - 4 x 2 9 + 1 2 1 2 - 1 2 ! - 4 x 2 9 2 + 1 2 1 2 - 1 1 2 - 2 3 ! - 4 x 2 9 3 + ⋯ ∞ = 1 - 2 x 2 9 - 1 2 · 1 2 2 · 16 x 4 81 - 1 2 · 1 2 · 3 2 6 · 64 x 6 729 + ⋯ ∞ = 1 - 2 x 2 9 - 2 x 4 81 - 8 x 6 729 + ⋯ ∞ ∴ 9 - 4 x 2 = 3 1 - 2 x 2 9 - 2 x 4 81 - 8 x 6 729 + ⋯ ∞