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Fungrim entry: 7212ea

n=0(1)nn+a=12(ψ ⁣(a+12)ψ ⁣(a2))\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:aC  and  a{0,1,}a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
\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\}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Infinity\infty Positive infinity
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    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)))))),
    Assumptions(And(Element(a, CC), NotElement(a, ZZLessEqual(0)))))

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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