A calculus of the absurd

$$2 + 2e^{\frac {\pi }{10}i} + 2e^{\frac {2\pi }{10}i} + 2e^{\frac {3\pi }{10}i} + 2e^{\frac {4\pi }{10}i}$$
forms a geometric series with $$a=2$$ and $$r=2e^{\frac {\pi }{10}i}$$. This means that the sum (as shown in the "sequences and series" section of these notes) is equal to
$$\frac {2\left ( 1 - \left (e^{\frac {\pi }{10}}\right )^5 \right )} {1-e^{\frac {\pi }{10}}}$$