I need to find the values of a ∈ R + for which the following sum converges. ∑ k = 1 ∞ a ∑ n = 1 k 1 n

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I need to find the values of   a  ∈            R        +   for which the following sum converges.      ∑          k      =      1        ∞        a                  ∑                  n          =          1                k                    1        n

Sawyer Day · Accepted Answer

We compute                                          lim                                k            →            +            ∞                          ⁡        k                  (                                                                      a                                                            ∑                                              n                        =                        1                                            k                                                                                      1                        n                                                                                                                                          a                                                            ∑                                              n                        =                        1                                                                    k                        +                        1                                                                                                            1                        n                                                                                                                  −            1                    )                =                              lim                                k            →            +            ∞                          ⁡        k                  (                                    a                              −                                  1                                      k                    +                    1                                                                        −            1                    )                =                              lim                                k            →            +            ∞                          ⁡        k                  (                                    e                              −                                  1                                      k                    +                    1                                                  log                ⁡                a                                      −            1                    )                =        −        log        ⁡        a        .            By Raabe&#039;s test the series converges when   −  log  ⁡  a  &amp;gt;  1, i.e.,   0  &amp;lt;  a  &amp;lt;      e          −      1       and diverges when   a  &amp;gt;      e          −      1      . To check the convergence for   a  =      e          −      1      , we use       ∑          n      =      1        k              1      n        &amp;lt;  log  ⁡  k  +  1, to find      ∑          k      =      1        ∞              e              −                  ∑                      n            =            1                    k                                      1            n                                ≥      e          −      1            ∑          k      =      1        ∞              e              −        log        ⁡        k              =      e          −      1            ∑          k      =      1        ∞              1      k        .But the harmonic series diverges, hence the series diverges when   a  =      e          −      1

I need to find the values of a ∈ <mrow class="MJX-TeXAtom-ORD"> <

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