What is the sum of the series $\displaystyle \frac{\pi}{e}-\frac{\pi}{e^2}+\frac{\pi}{e^3}-\frac{\pi}{e^4}+...+ (-1)^{n+1} \frac{\pi}{e^n} + ...$ ?

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The series is geometric with ratio $-\dfrac{1}{e}$. Since the absolute value is less than 1, the sum is given by:

$$S=\frac{a_1}{1-r}$$ $$S = \frac{\frac{\pi}{e}}{1-\left(-\frac{1}{e}\right)}$$ $$S = \frac{\pi}{e} \div \frac{e+1}{e}$$ $$S = \frac{\pi}{e} \times \frac{e}{e+1}$$ $$S = \boxed{ \dfrac{\pi}{e+1}}$$