<s>[INST] Expansion concerning the binomial theorem The question goes:
Expand $(1-2x)^{1/2}-(1-3x)^{2/3}$ as far as the 4th term.
Ans: $x + x^2/2 + 5x^3/6 + 41x^4/24$
How should I do it?
 [/INST] Substituting $r=1/2$ and $y=-2x$ in the binomial series
$$
(1 + y)^r  = 1 + \frac{r}{{1!}}y + \frac{{r(r - 1)}}{{2!}}y^2  + \frac{{r(r - 1)(r - 2)}}{{3!}}y^3  + \frac{{r(r - 1)(r - 2)(r - 3)}}{{4!}}y^4  +  \cdots 
$$
gives
$$
(1-2x)^{1/2} = 1 - x - x^2/2 -x^3/2 - (5/8)x^4 + \cdots,
$$
while substituting $r=2/3$ and $y=-3x$ gives
$$
(1-3x)^{2/3} = 1 - 2x - x^2  - (4/3)x^3  - (7/3)x^4  + \cdots.
$$
Now take the difference to get the desired result.
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