Find the first three non-zero terms in the Maclaurin series for $\mathrm{e}^x \sin(x+\pi)$.[3

Question

Find the first three non-zero terms in the Maclaurin series for $\mathrm{e}^x \sin(x+\pi)$.[3] It is given that the three terms found in part (a) are equal to the first three terms in the series expansion of $ax(1+bx)^c$ for small $x$, where $a$, $b$ and $c$ are constants. Find the exact values of $a$, $b$ and $c$. Use these values to find the coefficient of $x^4$ in the expansion of $ax(1+bx)^c$, giving your answer as a simplified rational number.[5]

Accepted Answer

(a) $-x-x^2-\dfrac{x^3}{3}+\cdots$. (b) $a=-1,b=\dfrac13,c=3$; coefficient of $x^4$ is $-\dfrac1{27}$.

2025 DHS P1 Q3

2025 DHS P1 Q3

Solution:

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