I have to find a sequence { f k } k = 1 ∞ , with f k : N → [ 0 , ∞ ), such that lim k → ∞ f k ( n ) = 0 for all n ∈ N and ∫ N f k d μ = 1 for all k ∈ N , where μ is the counting measure, i.e. ∑ n = 1 ∞ f k ( n ) = 1 for all k ∈ N .My main attempt was to write f k ( n ) = 1 k 2 n , what gives me lim k → ∞ f k ( n ) = 0 for all n ∈ N , but ∑ n = 1 ∞ f k ( n ) = 1 k ≠ 1 if k ≠ 1, then it is not valid. The two conditions seem to be incompatible.

Question

I have to find a sequence             {              f        k            }              k      =      1              ∞      , with       f    k    :      N    →  [  0  ,  ∞  ), such that       lim          k      →      ∞            f    k    (  n  )  =  0 for all   n  ∈      N   and       ∫                  N                  f    k    d  μ  =  1 for all   k  ∈      N  , where   μ is the counting measure, i.e.       ∑          n      =      1              ∞            f    k    (  n  )  =  1 for all   k  ∈      N  .My main attempt was to write       f    k    (  n  )  =      1          k              2        n            , what gives me       lim          k      →      ∞            f    k    (  n  )  =  0 for all   n  ∈      N  , but       ∑          n      =      1              ∞            f    k    (  n  )  =      1    k    ≠  1 if   k  ≠  1, then it is not valid. The two conditions seem to be incompatible.

iberistazi · Accepted Answer

One solution is      f    k    (  n  )  =  (      e    k    −  1  )      e          −      k      n      Then      lim          k      →      +      ∞            f    k    (  n  )  =  0and      ∑          n      =      1              +      ∞            f    k    (  n  )  =  (      e    k    −  1  )      ∑          n      =      1              +      ∞            e          −      k      n        =            (              e        k            −      1      )              e                  −          k                            1      −              e                  −          k                      =  1

I have to find a sequence <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false"

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