Fungrim home page

# Fungrim entry: 6547da

$\sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right)$
Assumptions:$r \in \mathbb{Z}_{\ge 1}$
References:
• https://doi.org/10.1080%2F10652469.2017.1376193
TeX:
\sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right)

r \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
DigammaFunctionZero$x_{n}$ Zero of the digamma function
Infinity$\infty$ Positive infinity
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Factorial$n !$ Factorial
ComplexLimit$\lim_{z \to a} f(z)$ Limiting value, complex variable
DigammaFunction$\psi\!\left(z\right)$ Digamma function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("6547da"),
Formula(Equal(Sum(Div(1, Pow(DigammaFunctionZero(n), Add(r, 1))), For(n, 0, Infinity)), Where(Div(ComplexDerivative(f(z), For(z, 0, r)), Factorial(r)), Equal(f(z), ComplexLimit(Parentheses(Sub(DigammaFunction(t), Div(DigammaFunction(t, 1), DigammaFunction(t)))), For(t, z)))))),
Variables(r),
Assumptions(Element(r, ZZGreaterEqual(1))),
References("https://doi.org/10.1080%2F10652469.2017.1376193"))

## 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