Using the standard series expansion, expand $\ln(1+\sin x)$ where $0 \le x

Question

Using the standard series expansion, expand $\ln(1+\sin x)$ where $0 \le x < \pi$, in ascending powers of $x$, up to and including the term in $x^3$.[3] The first two terms of the series in part (a) are equal to the first two terms in the series expansion of $\frac{ax}{(1+bx)^2}$, where $a$ and $b$ are non-zero constants. Find the values of $a$ and $b$. Hence write down the third term in the series expansion of $\frac{ax}{(1+bx)^2}$. [4]

Accepted Answer

(a) $x-\dfrac12x^2+\dfrac16x^3+O(x^4)$ (b) $a=1$, $b=\dfrac14$, third term $=\dfrac3{16}x^3$

2024 MI PRU2 P1 Q4

2024 MI PRU2 P1 Q4

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