n প্রাকৃতিক সংখ্যা 1,2,3,……, n থেকে পরপর দুটি সংখ্যা সরানো হয়। পুনরায় এর পাটিগণিত গড়...

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

lim_ (n rarr oo) [(1) / (n) + (n ^ (2)) / ((n + 1) ^ (3)) + (n ^ (2)) / ((n + 2) ^ (3)) + ...... + (1) / (8n)]

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

If n in N and Delta_ (n) = det [[n!, (N + 1)!, (N + 2)! (N + 1)!, (N + 2)!, (N + 3)! ( n + 2)!, (n + 3)!, (n + 4)!]] then lim_ (n rarr oo) ((3 ^ (n ^ (3)) - 5) Delta_ (n)) / (Delta_ (n + 1)) equals

a_ (n) = (1+ (1) / (n ^ (2))) (1+ (2 ^ (2)) / (n ^ (2))) ^ (2) (1+ (3 ^ ( 2)) / (n ^ (2))) ^ (3) ......... (1+ (n ^ (2)) / (n ^ (2))) ^ (n) then lim_ (n rarr oo) a_ (n) ^ (- (1) / (n ^ (2))) is equal to

lim_ (n rarr oo) (1.2 + 2.3 + 3.4 + .... + n (n + 1)) / (n ^ (3))

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

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

lim_ (n rarr oo) (1) / (n) [(1) / (n + 1) + (2) / (n + 2) + ... + (3n) / (4n)]