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Question

Let fp(β)=(cosβp2+isinβp2)+(cos2βp2+isin2βp2)(cosβ(p1)p2+isinβ(p1)p2)+(cosβp+isinβp)
then limn1/fn(π)=.

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Solution

fp(β)=cosβp2+isinβp2+...+(cosβp+isinβp)
=eiβp2+ei2βp2+ei3βp2+...+eipβp2
=eiβp2⎜ ⎜ ⎜1+eiβp2+ei2βp2+...+ei(p1)βp2⎟ ⎟ ⎟
=eiβp2⎜ ⎜ ⎜ ⎜ ⎜epβp21eβp21⎟ ⎟ ⎟ ⎟ ⎟
=eiβp2⎜ ⎜ ⎜ ⎜ ⎜eβp1eβp21⎟ ⎟ ⎟ ⎟ ⎟
limnfn(π)=limnei(πn2).eπn1eπn21 =limnei×0.πn2eπn2πn3eπn2 =limnn=

limn1fn(π)=0

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