Aperiodic Monotonic Divergent Series of Rational Numbers

HYPERSPECIES

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

$\displaystyle \sum_{\substack{n\geq 1 \\ m \text{ truncations} \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{n+m} = \sum_{\substack{n\geq{m+1} \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{n} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{m+1} + \underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{m+2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{m+3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{m+4} + \underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\gamma}_{\text{seed}} \,-\, \underbrace{H_m}_{\substack{\text{truncation} \\ \text{factor}}} \underbrace{-\,\ln(j+1)}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ m \text{ truncations} \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{2(n+m)} = \sum_{\substack{n\geq{m+1} \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{2n} = \underbrace{0+\cdots+0}_{k\text{ shifts}} + \dfrac{1}{2(m+1)} \\ + \underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{2(m+2)} + \underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{2(m+3)} + \underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{2}}_{\text{seed}} \,-\, \underbrace{\dfrac{H_m}{2}}_{\substack{\text{truncation} \\ \text{factor}}} \underbrace{-\,\dfrac{\ln(j+1)}{2}}_{\substack{\text{spacing factor}}}$ (semi-stable)

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

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{n} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\gamma}_{\text{seed}} \underbrace{-\ln(j+1)}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{2n} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{2}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{2n-1} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{2}+\ln 2}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{2}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{3n} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{9}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{12}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{3}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{3}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{3n-1} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{3}-\dfrac{\pi}{6\sqrt{3}}+\dfrac{\ln 3}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{3}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{3n-2} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{3}+\dfrac{\pi}{6\sqrt{3}}+\dfrac{\ln 3}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{3}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{4n} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{12}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{16}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{4n-1} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{11}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}-\dfrac{\pi}{8}+\dfrac{3\ln 2}{4}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{4n-2} = \dfrac{1}{2} \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{2n-1} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{10}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}+\dfrac{\ln 2}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{4n-3} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{9}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}+\dfrac{\pi}{8}+\dfrac{3\ln 2}{4}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{5n} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{10}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{15}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{20}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{5}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{5n-1} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{9}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}-\dfrac{\pi}{10}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}+\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{5n-2} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}-\dfrac{\pi}{10}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}-\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{5n-3} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+\dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}+\dfrac{\pi}{10}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}-\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{1}{5n-4} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}+\dfrac{\pi}{10}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}+\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

j-spaced monotonic divergent series of rational numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{n} = 1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\gamma}_{\text{seed}} \underbrace{-\ln(j+1)}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{2n} = \dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{2}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{2n-1} = 1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{2}+\ln 2}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{2}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{3n} = \dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{9}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{12}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{3}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{3}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{3n-1} = \dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{3}-\dfrac{\pi}{6\sqrt{3}}+\dfrac{\ln 3}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{3}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{3n-2} = 1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{3}+\dfrac{\pi}{6\sqrt{3}}+\dfrac{\ln 3}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{3}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{4n} = \dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{12}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{16}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{4n-1} = \dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{11}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}-\dfrac{\pi}{8}+\dfrac{3\ln 2}{4}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{4n-2} = \dfrac{1}{2} \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{2n-1} = \dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{10}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}+\dfrac{\ln 2}{2}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{4n-3} = 1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{9}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{4}+\dfrac{\pi}{8}+\dfrac{3\ln 2}{4}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{4}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{5n} = \dfrac{1}{5}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{10}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+ \\ \dfrac{1}{15}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{20}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{\gamma}{5}}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{5n-1} = \dfrac{1}{4}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{9}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}-\dfrac{\pi}{10}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}+\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{5n-2} = \dfrac{1}{3}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{8}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}-\dfrac{\pi}{10}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}-\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{5n-3} = \dfrac{1}{2}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{7}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}+\dfrac{\pi}{10}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}-\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing}}} \dfrac{1}{5n-4} = 1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\dfrac{1}{6}+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots \\ = \underbrace{\dfrac{\gamma}{5}+\dfrac{\pi}{10}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}+\dfrac{\ln{5}}{4}+\dfrac{1}{4\sqrt{5}}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right)}_{\text{seed}} \underbrace{-\dfrac{\ln(j+1)}{5}}_{\substack{\text{spacing factor}}}$ (semi-stable)

k-shifted monotonic divergent series of rational numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ k \text{ shifts}}} H_n = \underbrace{0+\cdots+0}_{k\text{ shifts}}+H_1+H_2+H_3+H_4+\cdots = \underbrace{\dfrac{1+\gamma}{2}}_{\text{seed}} \underbrace{-k\gamma}_{\substack{\text{shift} \\ \text{factor}}}$ (unstable)

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