Does this sequence of number converge? If yes, what is its limit?$$a_n=\sqrt[n]{\sum_{i=1}^n\cos^ni}$$We know that $\displaystyle\sqrt[n]{\sum_{i=1}^n\cos^2i}$ is much simpler. This is an advanced problem our teacher has given to us. I tried to check the values of each term and it seems that $\displaystyle\lim_{k\to\infty}a_{2k}=1$ and $\displaystyle\lim_{k\to\infty}a_{2k-1}=-1$, but some culculators show that $\displaystyle\lim_{n\to\infty}a_n=1$ (without a step-by-step solution though).
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