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Question
State, by writing first four terms, the expansion of the following, where |b| < |a|
`("a" + "b")^(-1/3)`
Solution
`("a" + "b")^(-1/3) = ["a"(1 + "b"/"a")]^(-1/3)`
= `"a"^(-1/3)(1 + "b"/"a")^(-1/3)`
= `"a"^(-1/3)[1 + (-1/3)("b"/"a") + ((-1/3)(-1/3 - 1))/(2!) ("b"/"a")^2 + ((-1/3)(-1/3 - 1)(-1/3 - 2))/(3!) ("b"/"a")^3 + ...]`
= `"a"^(-1/3)[1 - "b"/(3"a") + (-1/3)(-4/3)(1/2)("b"^2/"a"^2) + (-1/3)(-4/3)(-7/3)(1/6)("b"^3/"a"^3) + ...]`
= `"a"^(-1/3) [1 - "b"/(3"a") + (2"b"^2)/(9"a"^2) - (14"b"^3)/(81"a"^3) + ...]`
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