My textbook (RCA, Rudin) asserts (in the proof of theorem 5.15) that $$ \lim_{n\to\infty}\lVert{\sum_{k=-n}^n}e^{ikt}\rVert_1=\infty. $$ Why is this true?
I tried using Euler's formula to reduce the above to $$ \lim_{n\to\infty}\int_{0}^{\pi}\lvert 1+2\sum_{k=1}^n\cos(kt)\rvert~dt=\infty, $$ but I don't know how to show that either.