For n in N , consider the sequence { a n } defined by: a n = 1 n ∑ k = 1 n 1 k

Question

For n in       N  , consider the sequence       {          a              n              }   defined by:      a          n        =      1          n            ∑          k      =      1              n            1          k

Arturo Wallace · Accepted Answer

Using only elementary inequalities and no (improper) integral:For every       k    +    1    −      k    =            1                        k          +          1                +                  k                     and   2      k    &amp;lt;      k    +    1    +      k    &amp;lt;  2      k    +    1  , hence      k    +    1    −      k    &amp;lt;            1              2                  k                      &amp;lt;      k    −      k    −    1    .Summing the rightmost inequality yields      a    n    &amp;lt;      2          n            ∑          k      =      1        n        k    −      k    −    1    =  2.Likewise, summing the leftmost inequality yields      a    n    &amp;gt;      2          n            ∑          k      =      1        n        k    +    1    −      k    =  2                    n        +        1            −      1              n        &amp;gt;  2  −      2          n        .Since             2              n              →  0, this proves that       lim          n      →      ∞            a    n    =  2

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