`lim_(n-->oo) (1+(1/2)+(1/4) + ----+(1/2^n))/(1+(1/3)+(1/9)+------+(1/3^n))`

lim_ (n rarr oo) ((sqrt (n ^ (2) +1) + n) ^ (2)) / ((n ^ (6) +1) ^ (1/3))

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)))

The sum S = lim_ (n rarr oo) [(1) / (n + 1) + (1) / (n + 2) + (1) / (n + 3) + ----- + (1) / (2n)]

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

lim_ (n rarr oo) [(1) / (1.2) + (1) / (2.3) + (1) / (3.4) + ... + (1) / (n (n + 1))] =

lim_ (n rarr oo) (((n ^ (5) +2) ^ ((1) / (4))) - ((n ^ (2) +1) ^ ((1) / (3))) ) / ((n ^ (4) +2) ^ ((1) / (5)) - (n ^ (3) +1) ^ ((1) / (2))) is

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

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))