Sunday, March 3, 2019

sequences and series - Prove that the sumnx=1cos2(x) is bounded





I need to show that nx=1cos2(x) is bounded above. I know that there's a similar formula for nx=1sin(2x) but I can't seem to find it anywhere.


Answer



nk=1cos2(k)=nk=1cos(2k)+12=12(n+nk=1cos(2k))=12(n+Re(nk=1e2ik))=12(n+Re(e2ie2in1e2i1))



Note that |Re(e2ie2in1e2i1)|2|e2i1|, hence nk=1cos2(k)=n2+O(1)



This implies the sum is not bounded.


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