Explicit formula for the series ∑ k = 1 ∞ x k k ! ⋅ k

Question

Explicit formula for the series       ∑          k      =      1        ∞              x      k              k      !      ⋅      k

snowman8842 · Accepted Answer

You can have the closed form      ∑          k      =      1              ∞                  x      k              k      k      !        =  −  γ  −  ln  ⁡  (  −  x  )  −  Γ  (  0  ,  −  x  )  ,where   Γ  (  s  ,  x  ) is the upper incomplete gamma function. Another possible form is      ∑          k      =      1        ∞              x      k              k      !      ⋅      k        =  −  γ  −  ln  ⁡      (    −    x    )    −      E    i        (    1    ,    −    x    )    ,where  E  i  (  a  ,  z  )  =      ∫          1              ∞                  e              −        t        z                    t      a        d  t  ,    0  &amp;lt;  R  e  (  z  )  ,which is known as the exponential integral. The following relation between the exponential integral and the upper incomplete gamma function is useful  E  i  (  a  ,  z  )  =      z          a      −      1        Γ  (  1  −  a  ,  z  )  .

Explicit formula for the series sum_(k=1)^infty x^k/(k!* k)

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