Test the convergences of the following series ∑ n = 1 ∞ ( − 1 ) n sin ⁡ n n

Question

Test the convergences of the following series      ∑          n      =      1              ∞        (  −  1      )    n              sin      ⁡      n              n

Kalmukujobvg · Accepted Answer

Denote   θ  =  1  +  π,       a    n    =  (  −  1      )    n    sin  ⁡  n  =  sin  ⁡  (  n  θ  ) and   ϕ  (  x  )  =      1          x      . Then compute  A  (  x  )  =      ∑          1      ≤      k      ≤      x            a    k    =      Im        (          ∑              1        ≤        k        ≤        x                    e              i        k        θ              )    =      Im        (                            e                      i            (            ⌊            x            ⌋            +            1            )            θ                          −                  e                      i            θ                                                e                      i            θ                          −        1              )  Which yields      |    A  (  x  )      |    ≤      |                            e                      i            (            ⌊            x            ⌋            +            1            )            θ                          −                  e                      i            θ                                                e                      i            θ                          −        1              |    ≤      2                  |                    e                  i          θ                    −      1              |            So x↦A(x) is bounded. Using Abel&#039;s summation formula, you may write      ∑          1      ≤      n      ≤      x            a    n    ϕ  (  n  )  =  A  (  x  )  ϕ  (  x  )  −      ∫    1    x    A  (  x  )      ϕ    ′    (  x  )    d  x

Test the convergences of the following series sum_(n=1)^infty (-1)^n sin n/sqrt(n)

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