Fungrim entry: d02cf9

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

Assumptions:

r \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

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

r \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; 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
`Factorial`	$n !$	Factorial
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("d02cf9"),
    Formula(Equal(Sum(Div(Pow(-1, n), Pow(Add(n, a), r)), For(n, 0, Infinity)), Mul(Div(Pow(-1, r), Mul(Pow(2, r), Factorial(Sub(r, 1)))), Sub(DigammaFunction(Div(a, 2), Sub(r, 1)), DigammaFunction(Div(Add(a, 1), 2), Sub(r, 1)))))),
    Variables(r, a),
    Assumptions(And(Element(r, ZZGreaterEqual(2)), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

Topics using this entry

Digamma function