I am trying to prove that the series ∑ 1 ( m 1 2 + m 2 2 + ⋯ + m r 2 ) μ in which the summation extends over all positive and negative integral values and zero values of m 1 , m 2 , … , m r , except the set of simultaneous zero values, is absolutely convergent if μ &gt; r 2

Question

I am trying to prove that the series   ∑            1                        (                      m                          1                                      2                                +                      m                          2                                      2                                +          ⋯          +                      m                          r                                      2                                )                          μ                     in which the summation extends over all positive and negative integral values and zero values of       m    1    ,      m    2    ,  …  ,      m    r  , except the set of simultaneous zero values, is absolutely convergent if   μ  &amp;gt;            r      2

arhaitategr · Accepted Answer

Here is a way I like. We can rewrite your sum as      ∑          m      ∈                        Z                          m                    ∖      {      0      }            1          ‖      m              ‖                  2                          r          +          ϵ                    where   ϵ  &amp;gt;  0. Then since  ‖  x      ‖          2        ≥      max          i            |        x          i            |    ,by using the comparison test, we know that our original series will converge if      ∑          m      ∈                        Z                          m                    ∖      {      0      }            1                  max                  i                            |                    m                  i                                      |                          r          +          ϵ                    converges. Since the set of all m with   k  −  1  ≤      max          i            |        m          i            |    ≤  k has size   ≤  C      k          r      −      1       for some constant C (it is the surface of an r dimensional cube) we see that the above is bounded by      ∑          k      =      1              ∞        C            k              r        −        1                    k              r        +        ϵ              ≤  C      ∑          k      =      1              ∞            1          k              1        +        ϵ            which converges.

I am trying to prove that the series ∑ <mstyle displaystyle="true" scriptlevel

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New Questions in Calculus 2