I am trying to sum the series &#x2211;<!-- ∑ --> u <mrow class="MJX-TeXAtom-ORD">

Gretchen Schwartz

Gretchen Schwartz

Answered question

2022-06-29

I am trying to sum the series
u n
where
u n = a + n 1 j = 1 n ( a + j )
a > 0

Answer & Explanation

conveneau71

conveneau71

Beginner2022-06-30Added 17 answers

Since ( x ) n = Γ ( x + n ) Γ ( x ) your sum can be written as
k = 1 ( a + k 1 ) Γ ( a + 1 ) Γ ( a + k + 1 ) .
Note
a + k 1 Γ ( a + k + 1 ) = a + k ( a + k ) Γ ( a + k ) 1 Γ ( a + k + 1 ) = 1 Γ ( a + k ) 1 Γ ( a + k + 1 ) .
So
k = 1 ( a + k 1 ) Γ ( a + 1 ) Γ ( a + k + 1 ) = 1.

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