Expansion question involving Taylor or Binomial series?

Question

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I have tried to solve this question using binomial series, Taylor series but I seem to be heading nowhere… I seem to get lost every time I attempt it.

Answer 1

Using Taylor (which reduces to the binomial series for integer powers of a binomial and to the generalized binomial series for real powers), and limiting ourselves to the third order,

$$\left(1+\frac x2\right)^3-(1+3x)^{1/2}=1+\frac{3x}{2}+\frac{3x^2}{4}+\frac{x^3}{8}-\left(1+\frac{3x}2-\frac{9x^2}82+\frac{27x^3}{16}-\cdots\right)\\ =\frac{15x^2}8-\frac{25x^3}{16}+\cdots=\frac{15x^2}8\left(1-\frac{5x}6+\cdots\right)$$

and you are done.