Aperiodic Monotonic Divergent Series of Rational Numbers

HYPERSECTIONS

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

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{an+b} = -\dfrac{1}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,-\, \dfrac{1}{a}\,\ln(j+1) \\ = \dfrac{1}{a}\,\gamma - \dfrac{1}{a}\,H_{\frac{b}{a}} \,-\, \dfrac{1}{a}\,\ln(j+1)$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{a}{an+b} = -\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,-\, \ln(j+1) = \gamma \,-\, H_{\frac{b}{a}} \,-\, \ln(j+1)$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{n}{an+b} = -\dfrac{1}{2a} + \dfrac{b}{a^2}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) - \dfrac{k-j}{a(j+1)} + \dfrac{b}{a^2}\,\ln(j+1) \\ = -\dfrac{1}{2a} \,-\, \dfrac{b}{a^2}\,\gamma \,+\, \dfrac{b}{a^2}\,H_\frac{b}{a} \,-\, \dfrac{k-j}{a(j+1)} \,+\, \dfrac{b}{a^2}\,\ln(j+1)$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{an}{an+b} = -\dfrac{1}{2} \,+\, \dfrac{b}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,-\, \dfrac{k-j}{j+1} \,+\, \dfrac{b}{a}\,\ln(j+1) \\ = -\dfrac{1}{2} \,-\, \dfrac{b}{a}\,\gamma \,+\, \dfrac{b}{a}\,H_\frac{b}{a} \,-\, \dfrac{k-j}{j+1} \,+\, \dfrac{b}{a}\,\ln(j+1)$ (unstable)

j-spaced monotonic divergent series of rational numbers

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \dfrac{1}{an+b} = -\dfrac{1}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,-\, \dfrac{1}{a}\,\ln(j+1) \\ = \dfrac{1}{a}\,\gamma - \dfrac{1}{a}\,H_{\frac{b}{a}} \,-\, \dfrac{1}{a}\,\ln(j+1)$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \dfrac{a}{an+b} = -\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,-\, \ln(j+1) = \gamma \,-\, H_{\frac{b}{a}} \,-\, \ln(j+1)$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \dfrac{n}{an+b} = -\dfrac{1}{2a} + \dfrac{b}{a^2}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) + \dfrac{j}{a(j+1)} + \dfrac{b}{a^2}\,\ln(j+1) \\ = -\dfrac{1}{2a} \,-\, \dfrac{b}{a^2}\gamma \,+\, \dfrac{b}{a^2}\,H_\frac{b}{a} \,+\, \dfrac{j}{a(j+1)} \,+\, \dfrac{b}{a^2}\,\ln(j+1)$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \dfrac{an}{an+b} = -\dfrac{1}{2} \,+\, \dfrac{b}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,+\, \dfrac{j}{j+1} \,+\, \dfrac{b}{a}\,\ln(j+1) \\ = -\dfrac{1}{2} \,-\, \dfrac{b}{a}\,\gamma \,+\, \dfrac{b}{a}\,H_\frac{b}{a} \,+\, \dfrac{j}{j+1} \,+\, \dfrac{b}{a}\,\ln(j+1)$ (unstable)

k-shifted monotonic divergent series of rational numbers

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \dfrac{1}{an+b} = -\dfrac{1}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) = \dfrac{1}{a}\,\gamma \,-\, \dfrac{1}{a}\,H_{\frac{b}{a}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \dfrac{a}{an+b} = -\psi^{(0)}\!\left(1+\frac{b}{a}\right) = \gamma \,-\, H_{\frac{b}{a}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \dfrac{n}{an+b} = -\dfrac{1}{2a} + \dfrac{b}{a^2}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) - \dfrac{k}{a} \\ = -\dfrac{1}{2a} \,-\, \dfrac{b}{a^2}\,\gamma \,+\, \dfrac{b}{a^2}\,H_\frac{b}{a} \,-\, \dfrac{k}{a}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \dfrac{an}{an+b} = -\dfrac{1}{2} \,+\, \dfrac{b}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) \,-\, k \\ = -\dfrac{1}{2} \,-\, \dfrac{b}{a}\,\gamma \,+\, \dfrac{b}{a}\,H_\frac{b}{a} \,-\, k$ (unstable)

SECTIONS

$\displaystyle \sum_{n\geq0} \dfrac{1}{n+b} = -\psi^{(0)}(b) = \gamma - H_{b-1}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{n+b} = -\psi^{(0)}(1+b) = \gamma - H_{b}$ (semi-stable)

$\displaystyle \sum_{n\geq0} \dfrac{1}{n-b} = -\psi^{(0)}(-b) = \gamma - H_{-b-1} = \gamma - \dfrac{1}{2} - H_b$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{n-b} = -\psi^{(0)}(1-b) = \gamma - H_{-b} = \gamma - \dfrac{1}{2} - H_{b-1}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{an+b} = -\dfrac{1}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) = \dfrac{1}{a} \left(\gamma - H_{\frac{b}{a}}\right)$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{a}{an+b} = -\psi^{(0)}\!\left(1+\frac{b}{a}\right) = \gamma \,-\, H_{\frac{b}{a}}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{n}{an+b} = -\dfrac{1}{2a} \,+\, \dfrac{b}{a^2}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) = -\dfrac{1}{2a} \,-\, \dfrac{b}{a^2} \gamma \,+\, \dfrac{b}{a^2}\,H_\frac{b}{a}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{an}{an+b} = -\dfrac{1}{2} \,+\, \dfrac{b}{a}\,\psi^{(0)}\!\left(1+\frac{b}{a}\right) = -\dfrac{1}{2} \,-\, \dfrac{b}{a} \gamma \,+\, \dfrac{b}{a}\,H_\frac{b}{a}$ (semi-stable)

$\displaystyle \sum_{n\geq0} \dfrac{1}{n(n+b)} = \dfrac{1}{b} \left(H_{b-1}-\dfrac{1}{2}\right)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{1}{n(n+b)} = \dfrac{H_b}{b}$ (convergent)

$\displaystyle \sum_{n\geq0} \dfrac{1}{n(n-b)} = -\dfrac{H_b}{b}$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{1}{n(n-b)} = -\dfrac{H_{-b}}{b} = -\dfrac{1}{b} \left(H_{b-1}+\dfrac{1}{2}\right)$ (convergent)

$\displaystyle \sum_{n\geq0} \dfrac{1}{(n+a)(n+b)} = \dfrac{H_{a-1} - H_{b-1}}{a-b} \qquad (a\neq b)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{1}{(n+a)(n+b)} = \dfrac{H_a - H_b}{a-b} \qquad (a\neq b)$ (convergent)

$\displaystyle \sum_{n\geq0} \dfrac{1}{(n+a)(n-b)} = \dfrac{H_{a-1} - H_b - \frac{1}{2}}{a+b} \qquad (a+b \neq 0)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{1}{(n+a)(n-b)} = \dfrac{H_a - H_{b-1} - \frac{1}{2}}{a+b} \qquad (a+b \neq 0)$ (convergent)

$\displaystyle \sum_{n\geq0} \dfrac{1}{(n-a)(n-b)} = -\dfrac{H_{a} - H_{b}}{a-b} \qquad (a\neq b)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{1}{(n-a)(n-b)} = -\dfrac{H_{a-1} - H_{b-1}}{a-b} \qquad (a\neq b)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{1}{(an+b)(cn+d)} = \dfrac{H_\frac{b}{a} - H_\frac{d}{c}}{bc-ad} \qquad (bc \neq ad)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{en+f}{(an+b)(cn+d)} = \dfrac{e}{ac} \, \gamma \;+\; \left(f-\dfrac{be}{a}\right) \dfrac{H_\frac{b}{a}}{bc-ad} \;-\; \left(f-\dfrac{de}{c}\right) \dfrac{H_\frac{d}{c}}{bc-ad}$

where $(b\neq 0, d\neq 0 \text{ and } ad \neq bc)$ (convergent)

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