Solve the equation: \(\displaystyle{f{{\left({n}\right)}}}={{\cot}^{{2}}{\left({\frac{{\pi}}{{{n}}}}\right)}}+{{\cot}^{{2}}{\left({\frac{{{2}\pi}}{{{n}}}}\right)}}+\ldots+{{\cot}^{{2}}{\left({\frac{{{\left({n}-{1}\right)}\pi}}{{{n}}}}\right)}}\) then how to find limit

Lorelei Stanton

Lorelei Stanton

Answered question

2022-03-22

Solve the equation:
f(n)=cot2(πn)+cot2(2πn)++cot2((n1)πn)
then how to find limit of 3f(n)(n+1)(n+2) as n ?

Answer & Explanation

kattylouxlvc

kattylouxlvc

Beginner2022-03-23Added 11 answers

Recall the expansion:
tannx=(n1)tanx(n3)tan3x+1(n2)tan2x+(n4)tan4x+
=(n1)cotn1x(n3)cotn3x+cotn(n2)cotn2x+(n4)cotn4+
Now [kπn]k=1n1 are the roots of the equation: tannx=0
Thus, (n1)cotn1x(n3)cotn3x+=0 whenever, x=kπn for 1kn1
Thus, using Vieta's formula we might write:
k=1n1cot2kπn=(k=1n1cotkπn)221k1<k2n1cotk1πncotk2πn
=0+2(n3)(n1)
=(n1)(n2)3

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