1. **State the problem:** We need to find the sum of the infinite series $$\sum_{n=1}^\infty \frac{e^n}{\pi^n}$$ or determine if it diverges. 2. **Identify the type of series:** This is a geometric series with the general term $$a_n = \left(\frac{e}{\pi}\right)^n$$. 3. **Recall the formula for the sum of an infinite geometric series:** $$\text{If } |r| < 1, \quad \sum_{n=1}^\infty ar^{n-1} = \frac{a}{1-r}$$ Since our series starts at $n=1$, the sum is: $$\sum_{n=1}^\infty r^n = \frac{r}{1-r}$$ where $$r = \frac{e}{\pi}$$. 4. **Check the convergence condition:** Since $$e \approx 2.718$$ and $$\pi \approx 3.1415$$, we have $$\left|\frac{e}{\pi}\right| < 1$$ so the series converges. 5. **Calculate the sum:** $$\sum_{n=1}^\infty \left(\frac{e}{\pi}\right)^n = \frac{\frac{e}{\pi}}{1 - \frac{e}{\pi}} = \frac{\frac{e}{\pi}}{\frac{\pi - e}{\pi}} = \frac{e}{\pi - e}$$ 6. **Final answer:** The sum of the series is $$\boxed{\frac{e}{\pi - e}}$$ which corresponds to choice B.