Fungrim entry: 7212ea

\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n + a} = \frac{1}{2} \left(\psi\!\left(\frac{a + 1}{2}\right) - \psi\!\left(\frac{a}{2}\right)\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n + a} = \frac{1}{2} \left(\psi\!\left(\frac{a + 1}{2}\right) - \psi\!\left(\frac{a}{2}\right)\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("7212ea"),
    Formula(Equal(Sum(Div(Pow(-1, n), Add(n, a)), For(n, 0, Infinity)), Mul(Div(1, 2), Sub(DigammaFunction(Div(Add(a, 1), 2)), DigammaFunction(Div(a, 2)))))),
    Variables(a),
    Assumptions(And(Element(a, CC), NotElement(a, ZZLessEqual(0)))))

Topics using this entry

Digamma function