2022 FRQ

Part B

Question 6

The function \(f\) is defined by the power series for all real numbers \(x\) for which the series converges:

\[f(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots + \frac{(-1)^{n}x^{2n+1}}{2n+1} + \dots\]

1. Using the ratio test, find the interval of convergence of the power series for \(f\). Justify your answer.
2. Show that \(\left|f(\frac{1}{2})-\frac{1}{2}\right| \leq \frac{1}{10}\). Justify your answer.
3. Write the first four nonzero terms and the general term for an infinite series that represents \(f'(x)\).
4. Use the result from 3. to find the value of \(f'(\frac{1}{6})\).