find the expansion of $\frac{1}{{{\left( 1-3x \right)}^{3}}}$ in ascending powers of $x$ up to and i

Question

find the expansion of $\frac{1}{{{\left( 1-3x \right)}^{3}}}$ in ascending powers of $x$ up to and including the ${{x}^{2}}$ term. State the set of values of $x$ for which the series expansion is valid. Find, in terms of $n$, the coefficient of ${{x}^{n}}$ in the expansion in ascending powers of $x$ for $\frac{1}{{{\left( 1-3x \right)}^{3}}}$, leaving your answer in the simplest form.

Accepted Answer

(i) $1+9x+54x^2+\cdots$; (ii) $-\frac13<x<\frac13$; (iii) coefficient of $x^n$ is $\frac{(n+1)(n+2)}2 3^n$

NYJC P1 Q2

NYJC P1 Q2

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