Let a and b be positive reals. Show that lim n → ∞ ( a 1 n + b 1 n 2 ) n = a b

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Let a and b be positive reals. Show that      lim          n      →      ∞                  (                                    a                                          1                n                                              +                      b                                          1                n                                                    2            )        n    =      a    b

britspears523jp · Accepted Answer

An elementary proof. We use the Taylor series       e    x    =  1  +  x  +  O  (      x    2    ) and the fact that       lim          n      →      ∞        (  1  +  x      /    n      )    n    =      e    x  If   a  =  b the identity is trivial. Without loss of generality, assume   0  &amp;lt;  a  &amp;lt;  b. Then                                          (                                                            a                                      1                                          /                                        n                                                  +                                  b                                      1                                          /                                        n                                                              2                        )                    n                                    =                    b                              (                                          1                +                (                                  a                  b                                                  )                                      1                                          /                                        n                                                              2                        )                    n                                                  =                    b                              (                                          1                +                                  e                                                            1                      n                                        ln                    ⁡                                          a                      b                                                                                  2                        )                    n                                                  =                    b                              (            1            +                          1              2                                      1              n                        ln            ⁡                          a              b                        +            O            (            1                          /                                      n              2                        )            )                    n                                                  =                    b                              (            1            +                          1              n                        ln            ⁡                                          a                b                                      )                    n                +        O        (        1                  /                n        )        .            Therefore,                              lim                      n            →            ∞                                                (                                                            a                                      1                                          /                                        n                                                  +                                  b                                      1                                          /                                        n                                                              2                        )                    n                                    =                              lim                      n            →            ∞                          b                              (            1            +                          1              n                        ln            ⁡                                          a                b                                      )                    n                                                  =                    b                  e                      ln            ⁡                                          a                b                                                                                      =                              a          b                .

Kapalci · Accepted Answer

Another proof. Expand in a binomial series,            (                                    a                          1                              /                            n                                +                      b                          1                              /                            n                                      2            )        n    =      1          2      n            ∑          k      =      0        n                                (                            n        k                              )                      (      a          1              /            n            )    k    (      b          1              /            n            )          n      −      k        .Use the de Moivre-Laplace theorem,                              (                            n        k                              )                                (              1        2            )        k              (              1        2            )              n      −      k        (      a          1              /            n            )    k    (      b          1              /            n            )          n      −      k        ≃      1                  2        π            σ            e          −      (      k      −      μ              )        2                    /            (      2              σ        2            )        (      a          1              /            (      2      μ      )            )    k    (      b          1              /            (      2      μ      )            )          2      μ      −      k      where   μ  =  n      /    2 and       σ    2    =  n      /    4. Change variables. Let   z  =  (  k  −  μ  )      /    σ. Therefore,      lim          n      →      ∞                  (                                    a                          1                              /                            n                                +                      b                          1                              /                            n                                      2            )        n    =      lim          n      →      ∞            a    b        1          2      π            ∫          −      ∞        ∞    d  z        e          −              z        2                    /            2                  (              a        b            )              σ      z              /            (      2      μ      )        .The integral can be done easily enough by completing the square. We find      1          2      π            ∫          −      ∞        ∞    d  z        e          −              z        2                    /            2                  (              a        b            )              σ      z              /            (      2      μ      )        =  exp  ⁡                    σ        2                    log        2            ⁡      (      a              /            b      )              8              μ        2              .But   σ      /    μ  =  1      /        n  . Therefore, in the limit the integral is unity. Thus,      lim          n      →      ∞                  (                                    a                          1                              /                            n                                +                      b                          1                              /                            n                                      2            )        n    =      a    b    .

Let a and b be positive reals. Show that <munder> <mo form="prefix">lim <mrow class="MJX

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New Questions in Calculus 1