Using standard result given in the List of Formulae (MF26), find the Maclaurin’s series for \(\

Question

Using standard result given in the List of Formulae (MF26), find the Maclaurin’s series for $\ln (1+x+{{x}^{2}})$, up to and including the term in ${{x}^{3}}$.[2] Determine the set of values of $x$ for which the expansion in part (i) is valid.[3] Hence, or otherwise, obtain the series expansion of $\frac{1+2x}{1+x+{{x}^{2}}}$ up to and including the term in ${{x}^{2}}$.[2]

Accepted Answer

(i) $\displaystyle \ln(1+x+x^2)=x+\frac{x^2}{2}-\frac{2x^3}{3}+O(x^4)$. (ii) The expansion is valid for $\displaystyle -1\le x\le1$. (iii) $\displaystyle \frac{1+2x}{1+x+x^2}=1+x-2x^2+O(x^3)$, valid for $|x|<1$.

Maclaurin Series and Power Series - Substitution and Composition of Power Series

Maclaurin Series and Power Series - Substitution and Composition of Power Series

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