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Question
Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid
`(x + 2) - 2/3`
Solution
`(x + 2) - 2/3 = 1/(x + 2)^(2/3)`
= `1/(2^(2/3)(1 + x/2)^(2/3)`
= `2^((-2)/3)(1 + x/2)^(- 2/3)`
= `2^((-2)/3)(1 - 2/3(x/2) + (((-2)/3)((-2 - 1)/3)(x/2)^2)/(2!) ....)`
= `2^((-2)/3) (1 - x/3 + 10/18 x^2/4 - ....)`
= `2^((-2)/3){1 - x/5 + (5x^2)/36 - 5/81 x^3 ...}`
Hence `|x/2| < 1`
⇒ |x| < 2
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