Consider two geometric progressions `a_1,a_2,a_3,......a_0 & b_1, b_2, b_3,....b_n with a_r=1/b_r=2^(r-1) and an-`other sequence` t_1,t_2,t_3,......t_n`. such that `t_r = cot^-1 (2a_r + b_r)then lim_(n->oo)sum_(r=1)^nt_r` is -

The value of the lim_(n->oo)tan{sum_(r=1)^ntan^(- 1)(1/(2r^2))} is equal to

sum_(r=1)^n(2r+1)=...... .

If t_(1)=1,t_(r )-t_( r-1)=2^(r-1),r ge 2 , find sum_(r=1)^(n)t_(r ) .

If a_1, a_2, a_3,.....a_n are in H.P. and a_1 a_2+a_2 a_3+a_3 a_4+.......a_(n-1) a_n=ka_1 a_n , then k is equal to

Sum of the series sum_(r=1)^(n) (r^(2)+1)r! is

Let |Z_(r) - r| le r, Aar = 1,2,3….,n . Then |sum_(r=1)^(n)z_(r)| is less than

If sum_(r=1)^(n)T_(r)=(n(n+1)(n+2)(n+3))/(12) where T_(r) denotes the rth term of the series. Find lim_(nto oo) sum_(r=1)^(n)(1)/(T_(r)) .

The sequence a_(1),a_(2),a_(3),".......," is a geometric sequence with common ratio r. The sequence b_(1),b_(2),b_(3),".......," is also a geometric sequence. If b_(1)=1,b_(2)=root4(7)-root4(28)+1,a_(1)=root4(28)" and "sum_(n=1)^(oo)(1)/(a_(n))=sum_(n=1)^(oo)(1)/(b_(n)) , then the value of (1+r^(2)+r^(4)) is

Number of solutions of the equation cosectheta=k in [0,pi] where k=lim_(n->oo)pi_(r=2)^n((r^3-1)/(r^3+1)) is

Definite integration as the limit of a sum : lim_(ntooo)sum_(r=1)^(n)(1)/(n)e^(r/(n))=.............