Evaluate the Sum ∑ i = 0 ∞ i N 4 i

Question

Evaluate the Sum       ∑          i      =      0        ∞              i      N              4      i

x0xkatiex0xz6u · Accepted Answer

Note that for |x|&amp;lt;1, we have      ∑          i      =      0              ∞            x    i    =            1              1        −        x            We then have      ∑          i      =      0              ∞        i      x    i    =  x            d              d        x                  (          ∑              i        =        0                    ∞                    x      i        )    =  x            d              d        x                  (                  1                  1          −          x                      )  If we denote the operator   x            d              d        x             as L, we then have      ∑          i      =      0              ∞        i      x    i    =  L      (                  1                  1          −          x                      )        ∑          i      =      0              ∞            i    2        x    i    =  L      (    L          (                        1                      1            −            x                              )        )    =      L    2        (                  1                  1          −          x                      )  And in general,      ∑          i      =      0              ∞            i    N        x    i    =      L    N        (                  1                  1          −          x                      )  Hence,      ∑          i      =      0              ∞                          i        N                    4        i              =                          L        N                    (                              1                          1              −              x                                      )            |              x      =      1              /            4      Also, this is nothing but      Li          −      N            (                  1        4              )    ,where       Li    s    (  z  )  =            ∑              k        =        1                    ∞                                      z          k                          k          s                    , the Polylogarithm function.

Evaluate the Sum sum_(i=0)^infty i^N/4^i

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