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

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

lim_(n rarr oo)(sqrt(n+1)-sqrt(n))=0

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 U_(n)=(n!)/((n+2)!) where n in N. If S_(n)=sum_(n=1)^(n)U_(n), then lim_(n rarr oo)S_(n), equals

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)))=

Given S_(n)=sum_(k=1)^(n)(k)/((2n-2k+1)(2n-k+1)) and T_(n)=sum_(k=1)^(n)(1)/(k) then (T_(n))/(S_(n)) is equal to

Let S_(n)=sum_(r=1)^(oo)(1)/(n^(r)) and sum_(n=1)^(k)(n-1)S_(n)=5050, then k=

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

lim_(n rarr oo)(1)/(n)sum_(r=1)^(2n)(r)/(sqrt(n^(2)+r^(2))) equals