Finding lim n → ∞ ∫ 1 a n 1 + x n d x

Question

Finding       lim          n      →      ∞            ∫    1    a        n          1      +              x        n                d  x

barquegese2 · Accepted Answer

Answer:                              ∫          1          a                          n                      1            +                          x              n                                              d        x                            =                  ∫                      1                    a                          n                      x            n                                    (                      1                          1              +              (              1                              /                            x                              )                n                                              )                d        x                                          =                  ∫          1          a                          n                      x            n                                    (          1          −                      1                          x              n                                +                      1                          x                              2                n                                              −          ⋯          )                d        x                                          =        n                  ∫          1          a                          1                      x            n                          −                  1                      x                          2              n                                      +                  1                      x                          3              n                                      −        ⋯                d        x                                          =        n                  [                      1                          n              −              1                                −                      1                          2              n              −              1                                +          ⋯          ]                −                                                            n                                  [                                                            a                                              1                        −                        n                                                                                    n                      −                      1                                                        −                                                            a                                              1                        −                        2                        n                                                                                    2                      n                      −                      1                                                        +                  ⋯                  ]                                            ⏟                                            →                        0                                Since        lim          n      →      ∞            n          k      n      −      1        =      1    k    :      lim          n      →      ∞            ∫    1    a        n          1      +              x        n              d  x  =  1  −      1    2    +      1    3    −  ⋯  =  ln  ⁡  2

Finding lim_(n->infty) int_1^a n/(1+x^n)dx

Answered question

Answer & Explanation

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