By successively differentiating $(1+x)^n$, find the Maclaurin's series for $(1+x)^n$, up to

Question

By successively differentiating $(1+x)^n$, find the Maclaurin's series for $(1+x)^n$, up to and including the term in $x^3$.[3] Obtain the expansion of $(4-x)^{\frac{5}{2}}(1+2x^2)^{\frac{3}{2}}$ up to and including the term in $x^3$.[5] Find the set of values of $x$ for which the expansion in part (ii) is valid.[2]

Accepted Answer

(i) $(1+x)^n=1+nx+\frac{n(n-1)}2x^2+\frac{n(n-1)(n-2)}6x^3+\cdots$; (ii) $32-20x+\frac{399}{4}x^2-\frac{1925}{32}x^3+\cdots$; (iii) $-\frac1{\sqrt2}<x<\frac1{\sqrt2}$

2021 TGM P1 Q6

2021 TGM P1 Q6

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