[" on-negative integer "n," let "],[qquad f(n)=(sum_(k=0)^(n)sin((k+1)/(n+2)pi)sin((k+2)/(n+2)pi))/(sum_(k=0)^(n)sin^(2)((k+1)/(n+2)pi))]

For non-negative inger n, let f(n)=sum_(k=)^(n) sin((k+1)/(n+1)pisin((k+2)/(n+1)pi))/(sum_(k=0)^(n)sin^(2)((k+1)/(n+1)pi)) Assuming cos^(-1)x takes values in [0,pi] which of the following options is/are correct?

Let U_(n)=sum_(k=1)^(n)(n)/(n^(2)+k^(2)),S_(n)=sum_(k=0)^(n-1)(n)/(n^(2)+k^(2)) then

S_(n)=sum_(k=1)^(n)(k^(2)+n^(2))/(n^(3)) and T_(n)=sum_(k=0)^(n-1)(k^(2)+n^(2))/(n^(3)),n in N then

Let f(n)=(sum_(r=1)^(n)((1)/(r)))/(sum_(k=1)^(n)(k)/(2n-2k+1)(2n-k+1))

Let a_(n)=sum_(k=1)^(n)(1)/(k(n+1-k)), then for n>=2

sum_(m=1)^(4n+1)[sum_(K=1)^(m-1){sin((2k pi)/(m))-i cos((2k pi)/(m))}^(m)]

lim_(n->oo)sum_(k=1)^n((sin)pi/(2k)-(cos)pi/(2k)-(sin)(pi/(2(k+2))+(cos)pi/(2(k+2)))=

Evaluate lim_(ntooo)(1)/(n)[sin^(2k).(pi)/(2n)+sin^(2k).(2pi)/(2n)+....+sin^(2k).(pi)/(2)]

If n in N, sum_(k=1)^(n)cos^(-1)(x_(k))=npi then the value of sum_(k=1)^(n)sin^(-1)(x_(k))=