Expand ${{\left( 1+x \right)}^{8}}$ in ascending powers of $x$, up to and including the term in

Question

Expand ${{\left( 1+x \right)}^{8}}$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$. In the expansion of ${{(1+y+k{{y}^{2}})}^{8}}$ in ascending powers of $y$, the coefficient of ${{y}^{3}}$ is zero. Find the value of constant $k$. Given that $\left| x \right|<1$, expand $\frac{2x+3}{(x+1)(x+2)}$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$.

Accepted Answer

(a) $1+8x+28{{x}^{2}}+56{{x}^{3}}+...$, $k=-1$ (b) $\frac{3}{2}-\frac{5x}{4}+\frac{9{{x}^{2}}}{8}+...$

Maclaurin Series and Power Series

Maclaurin Series and Power Series