In the expansion of ${{(1+2x+3{{x}^{2}})}^{n}}$ in ascending powers of $x$, the coefficient of \

Question

In the expansion of ${{(1+2x+3{{x}^{2}})}^{n}}$ in ascending powers of $x$, the coefficient of ${{x}^{2}}$ is $105$. Given that $n$ is a positive integer, find the value of $n$. Obtain the expansion of $\frac{1-4x}{3+{{x}^{2}}}$ up to and including the term in ${{x}^{4}}$. Find the set of values of $x$ for which the expansion is valid.

Accepted Answer

(a) n=7; (b) (1/3)(1-4x-x^2/3+4x^3/3+x^4/9), valid for -sqrt(3)<x<sqrt(3).

Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series

Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series

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