Examine the uniform convergence of ∑ n = 1 ∞ x ( 1 + n x ) ( 1 + ( n + 1 ) x )

Question

Examine the uniform convergence of       ∑          n      =      1              ∞            x          (      1      +      n      x      )      (      1      +      (      n      +      1      )      x      )

Conor Frederick · Accepted Answer

Note that      x          (      1      +      n      x      )                        (                    1      +      (      n      +      1      )      x                        )                      =      1          1      +      n      x        −      1          1      +      (      n      +      1      )      x      and that therefore      S    m    (  x  )  =      1          1      +      x        −      1          1      +      (      m      +      1      )      x        .So, if  S  (  x  )  =      ∑          m      =      1        ∞        x          (      1      +      n      x      )                        (                    1      +      (      n      +      1      )      x                        )                      ,then  S  (  x  )  =      1          1      +      x        −      lim          n      →      ∞            1          1      +      n      x        =      {                                        1                          1              +              x                                                 if           x          &amp;gt;          0                                      0                           if           x          =          0.                        The convergence is not uniform on   [  0  ,  ∞  ), since  (  ∀  m  ∈      N    )  :      S    m        (          1              m        +        1              )    =            m      +      1              m      +      2        −      1    2  and      lim          m      →      ∞                  m      +      1              m      +      2        −      1    2    =      1    2    ≠  0  =  S  (  0  )  .Or you can say that if the convergence was uniform, then S would be continuous, but that&#039;s not the case.But the convergense is uniform on   [  a  ,  ∞  ), if a&amp;gt;0

Examine the uniform convergence of <munderover> ∑ <mrow class="MJX-TeXAt

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