Let `a_(n)=sum_(k=1)^(n)tan^(-1)((1)/(k^(2)+k+1));n>=1` ,then `lim_(n rarr oo)n^(2)(e^(a_(n)+1)-e^(a_(n)))=`

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

For n in N, let a_(n)=sum_(k=1)^(n)2k and b_(n)=sum_(k=1)^(n)(2k-1)* then lim_(n rarr oo)(sqrt(a)_(n)-sqrt(b_(n))) is equal to

If sum_(r=1)^(n)a_(r)=(1)/(6)n(n+1)(n+2) for all n>=1 then lim_(n rarr oo)sum_(r=1)^(n)(1)/(a_(r)) is

Let u_(n)=sum_(k=1)^(n)(k) and v_(n)=sum_(k=1)^(n)(k-0.5) . Then lim_(n rarr oo)(sqrt(u_(n))-sqrt(v_(n))) equals

COnsider the sequence (a_(n))n>=1, such that sum_(k=1)^(n)a_(k)=(3n^(2)+9n)/(2)AA n>=1. If lim_(n rarr oo)(1)/(na_(n))sum_(k=1)^(n)a_(k)=(p)/(q) (where p and q are coprime),then (p+q) is equal to

Let a sequence whose n^(th) term is {a_(n)} be defined as a_(1)=(1)/(2) and (n-1)a_(n-1)=(n+1)a_(n) for n>=2 then lim_(n rarr oo)S_(n), equals

If S_(n),=sum_(k=1)^(n)a_(k) and lim_(n rarr oo)a_(n)=a, then lim_(n rarr oo),(S_(n+1)-S_(n))/(sqrt(sum_(k=1)^(n)k)) is equal to

Let a_(1),a_(2),...,a_(n) .be sequence of real numbers with a_(n+1)=a_(n)+sqrt(1+a_(n)^(2)) and a_(0)=0 Prove that lim_(n rarr oo)((a_(n))/(2^(n-1)))=(2)/(pi)

Define a sequence (a_(n)) by a_(1)=5,a_(n)=a_(1)a_(2)…a_(n-1)+4 for ngt1 . Then lim_(nrarroo) (sqrt(a_(n)))/(a_(n-1))