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

Angel Kline

Angel Kline

Answered question

2022-10-31

Explicit formula for the series k = 1 x k k ! k

Answer & Explanation

snowman8842

snowman8842

Beginner2022-11-01Added 12 answers

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 < 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 ) .

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