Aperiodic Alternate Divergent Series of Rational Numbers

HYPERSECTIONS

j-spaced and k-shifted alternate divergent series of rational numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{(-1)^{n-1}}{an+b} = \dfrac{1}{a} \left(\ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2a}\right) - \psi^{(0)}\!\left(1+\dfrac{b}{a}\right)\right) \\ = \dfrac{1}{a} \left(\ln 2 + H_\frac{b}{2a} - H_\frac{b}{a}\right)$ (convergent)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{(-1)^{n-1}}{n+b} = \ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2}\right) - \psi^{(0)}\!\left(1+b\right) \\ = \ln 2 + H_\frac{b}{2} - H_b$ (convergent)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{(-1)^{n-1}}{an} = \dfrac{\ln 2}{a}$ (convergent)

SUPERSECTIONS

SECTIONS

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} \dfrac{an}{an+b} = \dfrac{1}{2} \,-\, \dfrac{b}{a} \left(\ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2a}\right) - \psi^{(0)}\!\left(1+\dfrac{b}{a}\right)\right) \\ = \dfrac{1}{2} \,-\, \dfrac{b}{a} \left(\ln 2 + H_\frac{b}{2a} - H_\frac{b}{a}\right)$ (stable)

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} \dfrac{n}{an+b} = \dfrac{1}{2a} \,-\, \dfrac{b}{a^2} \left(\ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2a}\right) - \psi^{(0)}\!\left(1+\dfrac{b}{a}\right)\right) \\ = \dfrac{1}{2a} \,-\, \dfrac{b}{a^2} \left(\ln 2 + H_\frac{b}{2a} - H_\frac{b}{a}\right)$ (stable)

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} \dfrac{n}{n+b} = \dfrac{1}{2} \,-\, b \left(\ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2}\right) - \psi^{(0)}(1+b)\right) \\ = \dfrac{1}{2} \,-\, b \left(\ln 2 + H_\frac{b}{2} - H_b\right)$ (stable)

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} \dfrac{1}{an+b} = \dfrac{1}{a} \left(\ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2a}\right) - \psi^{(0)}\!\left(1+\dfrac{b}{a}\right)\right) \\ = \dfrac{1}{a} \left(\ln 2 + H_\frac{b}{2a} - H_\frac{b}{a}\right)$ (convergent)

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} \dfrac{1}{n+b} = \ln 2 + \psi^{(0)}\!\left(1+\dfrac{b}{2}\right) - \psi^{(0)}(1+b) \\ = \ln 2 + H_\frac{b}{2} - H_b$ (convergent)

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} \dfrac{1}{an} = \dfrac{\ln 2}{a}$ (convergent)

SUBSECTIONS

INFRASECTIONS

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