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) $1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{6}x^3 + \dots$; (ii) $8 - 3x + \frac{387}{16}x^2 - \frac{1151}{128}x^3 + \dots$; (iii) $-\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$.

Using standard series

Using standard series