Question

Calculate the reciprocal of the coefficient of \(z^3\) in the Taylor series expansion of the function \(f(z)=\sin(z)\) around \(z=0\). (Provide the answer as an integer.)

Hint:

- Memorize Maclaurin series of basic functions (\(\sin z,\ \cos z,\ e^z\))—they save time in coefficient questions.

Answer 1

Correct Answer: -6

Step 1: The Taylor series of \(\sin z\) about \(z=0\) is \[ \sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots \] Hence the coefficient of \(z^3\) is \(-\frac{1}{3!}=-\frac{1}{6}\). Its reciprocal is \[ \left(-\frac{1}{6}\right)^{-1} = -6. \]