The problem is to find the sum of the infinite series $\sum_{k=1}^{\infty} (\frac{e}{\pi})^{k+1}$.

1. Problem Description

The problem is to find the sum of the infinite series $\sum_{k=1}^{\infty} (\frac{e}{\pi})^{k+1}$.

2. Solution Steps

The given series is a geometric series. We can rewrite the series as

$\sum_{k=1}^{\infty} (\frac{e}{\pi})^{k+1} = \sum_{k=1}^{\infty} (\frac{e}{\pi})(\frac{e}{\pi})^k = \frac{e}{\pi} \sum_{k=1}^{\infty} (\frac{e}{\pi})^k$.

The sum of an infinite geometric series $\sum_{k=1}^{\infty} ar^{k-1}$ is $\frac{a}{1-r}$ if $|r| < 1$. Alternatively, we can write $\sum_{k=1}^\infty r^k = \frac{r}{1-r}$ if $|r| < 1$.

In our case, we have $\sum_{k=1}^{\infty} (\frac{e}{\pi})^k$. Here, $r = \frac{e}{\pi}$. Since $e \approx 2.718$ and $\pi \approx 3.141$, we have $\frac{e}{\pi} < 1$. Thus, the series converges.

Using the formula for the sum of an infinite geometric series, we have

$\sum_{k=1}^{\infty} (\frac{e}{\pi})^k = \frac{\frac{e}{\pi}}{1 - \frac{e}{\pi}} = \frac{\frac{e}{\pi}}{\frac{\pi - e}{\pi}} = \frac{e}{\pi - e}$.

Therefore, the sum of the given series is

$\frac{e}{\pi} \sum_{k=1}^{\infty} (\frac{e}{\pi})^k = \frac{e}{\pi} \cdot \frac{e}{\pi - e} = \frac{e^2}{\pi(\pi - e)}$.

3. Final Answer

$\frac{e^2}{\pi(\pi - e)}$