Sunday, May 29, 2016

trigonometry - Prove that series with (1)kcos3(k) has sequence of partial sums bounded.



I found that (1)kcos3(k) equals (1sin2(k))cos((π+1)k) or cos((π+3)k)+sin(k)sin((π+2)k)+sin(k)cos(k)sin((π+1)k).



I wanted to flatten it into such a sum of terms that each has bounded partial sums, so the final sequence of partial sums is bounded too.



But I got the sin(k) and sin(k)cos(k).




I should use that (1)k equals cos(πk) and the trigonometric identities.


Answer



|n1k=0(1)kcos3(k)|=|n1k=0(1)k8(eik+eik)3|=|n1k=0(1)k8(e3ik+3eik+3eik+e3ik)|18|1(e3i)n1+e3i|+38|1(ei)n1+ei|+38|1(ei)n1+ei|+18|1(e3i)n1+e3i|14cos(32)+34cos(12)


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