Date modified: Wed 2023-08-02 . 10:54 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Extra credit problem: Find the exact values of these numbers. Warm up. Find the exact value (not decimal approximations) of the following numbers. Show your work. 1. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}\left( \frac{2022}{2023} \right)^{n}$ 2. $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2023)^{2n+1}(2n+1)}$ The answers should all be in familiar functions. And use the techniques shown in class to find them. Ok good. Now try this. 1. By completing the square, show that $$\int_{0}^{1/2} \frac{dx}{x^{2}-x+1}=\frac{\pi}{3 \sqrt{3}}$$ 2. By factoring $x^{3}+1$ as a sum of cubes, rewrite the integral in part (1) above. Then express $\displaystyle\frac{1}{1+x^{3}}$ as a powerseries and prove the following formula for $\pi$ : $$ \pi = \frac{3\sqrt{3}}{4} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{8^{n}}\left( \frac{2}{3n+1} +\frac{1}{3n+2} \right) $$ ///