Does the limit lim n → ∞ ∑ k = 1 n ln ⁡ ( k ( 1 + α ) n − k + 1 + ( n − k ) α ) exist?

Question

Does the limit      lim          n      →      ∞            ∑          k      =      1        n    ln  ⁡      (                  k        (        1        +        α        )                    n        −        k        +        1        +        (        n        −        k        )        α              )  exist?

Jake Mcpherson · Accepted Answer

Let   1  +  α  =      β          −      1      . We can rewrite the term within the logarithm as  f  (  n  ,  k  ,  β  )  =      k          n      −      k      +      β      We can find the finite sum      S    n    (  β  )  =      ∑          k      =      1        n    ln  ⁡  f  (  n  ,  k  ,  β  )  =  ln  ⁡      [          ∏              k        =        1            n        f    (    n    ,    k    ,    β    )    ]  Write out a few terms of the product      ∏          k      =      1        n    f  (  n  ,  k  ,  β  )  =      1          n      +      β      −      1        ⋅      2          n      +      β      −      2        ⋯      n    β  Which may be written as      ∏          k      =      1        n    f  (  n  ,  k  ,  β  )  =            Γ      (      n      +      1      )              Γ      (      n      +      β      )              /            Γ      (      β      )      So we have for the sum      S    n    (  β  )  =  ln  ⁡      [                  Γ        (        n        +        1        )        Γ        (        β        )                    Γ        (        n        +        β        )              ]  Using the asymptotic expansion of   Γ for large argument with fixed   β      S    n    (  β  )  ∼  (  1  −  β  )  ln  ⁡  n    ,    n  →  ∞So that the sum in logarithmically divergent unless   β  =  1, in which case       S    n    (  1  )  =  ln  ⁡  Γ  (  1  )  =  0

Does the limit <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXA

Answered question

Answer & Explanation

New Questions in Calculus 1