The Maclaurin series for $\arctan(x)$ is $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1}$ with $R = 1$. Using this series, find the exact value of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$.