I need to show that n∑x=1cos2(x) is bounded above. I know that there's a similar formula for n∑x=1sin(2x) but I can't seem to find it anywhere.
Answer
n∑k=1cos2(k)=n∑k=1cos(2k)+12=12(n+n∑k=1cos(2k))=12(n+Re(n∑k=1e2ik))=12(n+Re(e2ie2in−1e2i−1))
Note that |Re(e2ie2in−1e2i−1)|≤2|e2i−1|, hence n∑k=1cos2(k)=n2+O(1)
This implies the sum is not bounded.
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