# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC