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

Question

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

Accepted Answer

(i) $1+nx+\frac{n(n-1)}2x^2+\frac{n(n-1)(n-2)}6x^3+O(x^4)$. (ii) $8-3x+\frac{387}{16}x^2-\frac{1151}{128}x^3+O(x^4)$. (iii) Standard binomial condition: $-\frac1{\sqrt2}<x<\frac1{\sqrt2}$. The full series also converges at both endpoints, giving $\left[-\frac1{\sqrt2},\frac1{\sqrt2}\right]$.

Using standard series

Using standard series

Solution:

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