How do I find lim n → ∞ n ∫ 0 1 x n x 2 + 1 d x ?

Question

How do I find         lim          n      →      ∞        n      ∫          0              1                  x      n                      x        2            +      1        d  x   ?

Crevani9a · Accepted Answer

I              n        +        2              +          I      n        =          1              n        +        1            Let first define             I              (        n        →        ∞        )              =    I  Here, I believe that there shouldn&#039;t be any difference for a value       I    n   and       I          n      +      2       as   n  →  ∞ they must be same because n is already a very very big number then any small increment like(2) shouldn&#039;t make difference provided n is very big number.                              I                      n            +            2                          +                  I          n                                    =                  1                      n            +            1                                                                      lim                      n            →            ∞                          n                  I                      n            +            2                          +                  lim                      n            →            ∞                          n                  I          n                =                  lim                      n            →            ∞                                    n                      n            +            1                                                              Now, By intuition we can imagine this that when                       n            →            ∞                     then                                     I                              n                +                2                                      =                          I              n                                 = I                                          (As, from above blue color                   I                      (            n            →            ∞            )                          =        I        )                                  2        ×                  lim                      n            →            ∞                          n        I        =                  lim                      n            →            ∞                                    n                      n            +            1                          =        1                                                  ⟹                          lim                      n            →            ∞                          n        I        =                  1          2

Rachel Villa · Accepted Answer

We know       x    n    ≥      x          n      +      2       in the range   [  0  ,  1  ], so that       I    n    ≥      I          n      +      2      , implying      I    n    ≥      1    2        1          n      +      1        .Similarly      I    n    ≤      1    2        1          n      −      1        .The resulting upper and lower bounds for   n      I    n   each approach 1/2.

How do I find <mspace width="thickmathspace" /> <munder> <mo movablelimits="true" form="pre

Answered question

Answer & Explanation

New Questions in Calculus 1