Analyze the convergence behavior of the following series: ∑ k = 0 ∞ x 2 k 2 2 k − x 2 k + 1 3 2 k + 1 .

Question

Analyze the convergence behavior of the following series:      ∑          k      =      0              ∞                  x              2        k                    2              2        k              −            x              2        k        +        1                    3              2        k        +        1              .

cuprins60 · Accepted Answer

I assume that      ∑          k      =      0              ∞            (                  x                  2          k                            2                  2          k                      −                  x                  2          k          +          1                            3                  2          k          +          1                      )  is intended. You can avoid worrying about uniform convergence by using the ratio test. Calling the k-th term       a    k  , and letting   u  =  x      /    2, we have      a    k    =            (              x        2            )              2      k        −            (              x        3            )              2      k      +      1        =      u          2      k        −            (              2        3            )              2      k      +      1            u          2      k      +      1        =      u          2      k            (    1    −                  (                  2          3                )                    2        k        +        1              u    )      ,so      lim          k      →      ∞            |                  a                  k          +          1                            a        k              |    =      lim          k      →      ∞            u    2        |                  1        −        (        2                  /                3                  )                      2            k            +            3                          u                    1        −        (        2                  /                3                  )                      2            k            +            1                          u              |    =      u    2      .Clearly you need       u    2    &amp;lt;  1, or       |    x      |    &amp;lt;  2, and it’s not hard to check that you don’t get convergence at either endpoint.

Analyze the convergence behavior of the following series: <munderover> ∑

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