Fungrim entry: 19e67f

\sum_{n=A}^{B} \frac{1}{n + a} = \psi\!\left(a + B + 1\right) - \psi\!\left(a + A\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; A \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; B \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; A \le B \;\mathbin{\operatorname{and}}\; -a \notin \{A, A + 1, \ldots, B\}

TeX:

\sum_{n=A}^{B} \frac{1}{n + a} = \psi\!\left(a + B + 1\right) - \psi\!\left(a + A\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; A \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; B \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; A \le B \;\mathbin{\operatorname{and}}\; -a \notin \{A, A + 1, \ldots, B\}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("19e67f"),
    Formula(Equal(Sum(Div(1, Add(n, a)), For(n, A, B)), Sub(DigammaFunction(Add(Add(a, B), 1)), DigammaFunction(Add(a, A))))),
    Variables(a, A, B),
    Assumptions(And(Element(a, CC), Element(A, ZZ), Element(B, ZZ), LessEqual(A, B), NotElement(Neg(a), Range(A, B)))))

Topics using this entry

Digamma function