`1 + 1/2 + 1/2^2 + ...+ 1/ 2^(n-1) = `

If S_(n)=1 + (1)/(2) + (1)/(2) + …..+ (1)/(2^(n-1)), (n in N) then …….

Let S_n=1/1^2 + 1/2^2 + 1/3^2 +….. + 1/n^2 and T_n=2 -1/n , then :

Let S_n=1/1^2 + 1/2^2 + 1/3^2 +….. + 1/n^2 and T_n=2 -1/n , then :

lim_ (n rarr oo) (1+ (1) / (2) + (1) / (2 ^ (2)) + (1) / (2 ^ (3)) + ...... (1) / (2 ^ (n))) / (1+ (1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) ...... (1) / (3 ^ (n)))

lim_ (n rarr oo) n ^ (- n ^ (2)) ((n + 1) (n + (1) / (2)) (n + (1) / (2 ^ (2))) .... (n + (1) / (2 ^ (n-1)))) ^ (n)

lim_ (n rarr oo) [(1) / (1-n ^ (2)) + (2) / (1-n ^ (2)) + ... + (n) / (1-n ^ (2 ))] is

If a 1 , a 2 , a 3 , , a 2 n + 1 are in A.P., then a 2 n + 1 − a 1 a 2 n + 1 + a 1 + a 2 n − a 2 a 2 n + a 2 + + a n + 2 − a n a n + 2 + a n is equal to a. n ( n + 1 ) 2 × a 2 − a 1 a n + 1 b. n ( n + 1 ) 2 c. ( n + 1 ) ( a 2 − a 1 ) d. none of these

lim_ (n rarr oo) [n ^ ((1) / (2)) + (n + 1) ^ ((1) / (2))] ^ (- 1) -: n ^ (- (1) / (2))

If S_n=sum_(r=1)^n(1+2+2^2+ .......+2^r)/(2^r), then S_n is equal to (a) 2^n n-1 (b) 1-1/(2^n) (c) n -1+1/(2^n) (d) 2^n-1

lim_ (n rarr oo) [(1+ (1) / (n ^ (2)))) (1+ (2 ^ (2)) / (n ^ (2))) (1+ (3 ^ (2) ) / (n ^ (2))) ...... (1+ (n ^ (2)) / (n ^ (2)))] ^ ((1) / (n))