How to determine that this series is conditionally convergent? ∑ n = 1 ∞ ∫ n n + 1 sin ⁡ π x x p + 1 d x , 0 &lt; p ≤ 1

Question

How to determine that this series is conditionally convergent?      ∑          n      =      1        ∞        ∫    n          n      +      1                  sin      ⁡      π      x                      x        p            +      1              d    x  ,    0  &amp;lt;  p  ≤  1

Mya Hurst · Accepted Answer

Notice that   sin  ⁡  (  π  x  ) doesn&#039;t change its sign on the interval   [  n  ,  n  +  1  ] since its zeros are   x  =  k  ∈      Z  . By the mean value theorem there&#039;s a       c    n    ∈  [  n  ,  n  +  1  ] such that      ∫    n          n      +      1                  sin      ⁡      (      π      x      )                      x        p            +      1          d  x  =      1          (              c        n                    )        p            +      1            ∫    n          n      +      1        sin  ⁡  (  π  x  )    d  xFor the last integral, we have      ∫    n          n      +      1        sin  ⁡  (  π  x  )    d  x  =  −      1    π    cos  ⁡  (  π  x  )                    |              n          n      +      1          =  −      1    π        (    cos    ⁡    (    (    n    +    1    )    π    )    −    cos    ⁡    (    n    π    )    )    =            2      ⋅      (      −      1              )        n              π  so you have to establish the convergence of      2    π        ∑          n      =      1              ∞                  (      −      1              )        n                    (              c        n                    )        p            +      1      which follows from the alternating series test, and the divergence of      2    π        ∑          n      =      1              ∞            1          (              c        n                    )        p            +      1      which follows from a comparison with the divergent series       2    π        ∑          n      =      1              ∞            1          (      n      +      1              )        p            +      1        =      2    π        ∑          n      =      2              ∞            1                  n        p            +      1

How to determine that this series is conditionally convergent? <munderover> ∑<!-- ∑ -

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