I would like to prove convergence of the p-harmonic series 1 + 1 2 p + 1 3 p + 1 4 p + …

Question

I would like to prove convergence of the p-harmonic series  1  +      1          2      p        +      1          3      p        +      1          4      p        +  …

Samuel Vang · Accepted Answer

Let us prove that the sequence       S    n    =      ∑          k      =      1        n        1          k      2       is a Cauchy sequence. Let   ε  &amp;gt;  0 and we will define   N  ∈      N   as   N  =      ⌈          1      ε        ⌉  . Than for all   m  &amp;gt;  n  &amp;gt;  N we can observe that                              |                      S            n                    −                      S            m                    |                =                              |                      ∑                          k              =              n              +              1                        m                                1                          k              2                                |                =                  ∑                      k            =            n            +            1                    m                          1                      k            2                          ≤                  ∑                      k            =            n            +            1                    m                          1                      k            (            k            −            1            )                          =                                          =                                      ∑                      k            =            n            +            1                    m                          1                      k            −            1                          −                  1          k                =                  1          n                −                  1          m                ≤                  1          n                &amp;lt;                  1          N                &amp;lt;        ε            as wanted.

Jazmine Bruce · Accepted Answer

For p=2 we can do this:      1          n      2        &amp;lt;      1          n      (      n      −      1      )        =      1          n      −      1        −      1    n  and use a finite telescoping sum to prove the Cauchy criterion. For   1  &amp;lt;  m  &amp;lt;  n      S    n    −      S    m    =      ∑          k      =      m      +      1        n        1          k      2        &amp;lt;      ∑          k      =      m      +      1        n        (          1              k        −        1              −          1      k        )    =      1    m    −      1    n    &amp;lt;      1    m    .

I would like to prove convergence of the p-harmonic series 1 + 1 2 p

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New Questions in Calculus 2