Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist?

Question

Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist? If yes, what is its value?

Answer 1

Think about it. It is the length of a unit circle. That is 2$\pi$

Answer 2

This is not a rigorous argument but still will give you some intuition. Note that $$\sum_{k=1}^n \left \vert \exp \left(\dfrac{2 \pi i k}n - \dfrac{2 \pi i (k-1)}n\right)\right \vert$$ gives the perimeter of the $n$ sided polygon. So...