Let `S_n=1+2+3++n` and `P_n=(S_2)/(S_2-1)*(S_3)/(S_3-1)*(S_4)/(S_4-1)* . . . *(S_n)/(S_n-1)` Where `n in N ,(ngeq2)dot` Then `lim_(n→oo)P_n=______`

Let S_(n)=cot^(-1)2 +cot^(-1)8 +cot^(-1)18 +cot^(-1)32+…. to n^(th) term. Then lim_(n to oo)S_(n) is

If S_(n)=(1^(2)-1+1)(1!)+(2^(2)-2+1)(2!)+...+(n^(2)-n+1)(n!) , then S_(50)=

If S_n=cot^(-1)(3)+cot^(-1)(7)+cot^(-1)(13)+cot^(-1)(21)+....n terms, then

If S_(1), S_(2), S_(3),…., S_(n) are the sums to infinity of n infinte geometric series whose first terms are 1,2,3,… n and whose common ratios are (1)/(2), (1)/(3), (1)/(4), ….(1)/(n+1) respectively, show that, S_(1) + S_(2) + S_(3) +…S_(n) = (1)/(2)n(n+3)

If S=a_1+a_2+......+a_n,a_i in R^+ for i=1 to n, then prove that S/(S-a_1)+S/(S-a_2)+......+S/(S-a_n) ge n^2/(n-1), AA n ge 2

If S_(1), S_(2), S_(3),...S_(n) are the sums of infinite geometric series, whose first terms are 1,2,3,...n whose ratios are (1)/(2),(1)/(3),(1)/(4) ,...(1)/(n+1) respectively, then find the value of S_(1)^(2)+S_(2)^(2)+S_(3)^(2)+...+S_(2n-1)^(2) .

The absolute value of the sum of first 20 terms of series, if S_(n)=(n+1)/(2) and (T_(n-1))/(T_(n))=(1)/(n^(2))-1 , where n is odd, given S_(n) and T_(n) denotes sum of first n terms and n^(th) terms of the series

The sum S_(n)=sum_(k=0)^(n)(-1)^(k)*^(3n)C_(k) , where n=1,2,…. is

The value of lim_(n->oo)[(2n)/(2n^2-1)cos((n+1)/(2n-1))-n/(1-2n)dot(n(-1)^n)/(n^2+1)]i s (a) 1 (b) -1 (c) 0 (d) none of these