Question

please ans without l hospital rule limit class 12th ​

Answer 1

Step-by-step explanation:

α = lim (n->inf) [ (1³-1²) + (2³-2²)+(3³-3²)+...+(n³-n²)]/n^4

=> α = lim(n->inf) [ sigma(n=1 to n=inf) (n³-n²)/n^4 ]

=> α = (sigma(n=1 to inf) lim(n->inf) (n³/n^4 - n²/n^4)

=> α = (sigma(n=1 to inf) lim(n->inf) (1/n - 1/n²)

=> α = (sigma(n=1 to inf) ( 1/infi - 1/inf²)

=> α = (sigma(n=1 to inf) (0-0)

=> α = 0

Answer 2

Answer :

1/4

Solution:

Please refer to the attachments .

Note:

★ Sum of n natural numbers is given by ;

1 + 2 + 3 + . . .+ n = (n/2)•(n + 1)

★ Sum of squares of n natural numbers is given by ;

1² + 2² + 3² + . . . + n² = (n/6)•(n + 1)•(2n + 1)

★ Sum of cubes of n natural numbers is given by ;

1³ + 2³ + 3³ + . . . + n³ = [ (n/2)•(n + 1) ]²