Monotonic Divergent Series of Real Numbers

HYPERSECTIONS

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

$\displaystyle \sum_{\substack{n\geq m+1 \\ j \text{ spacing} \\ k \text{ shifts}}} \ln(an+b) = \dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2} \,-\, \ln\!\Gamma\!\left(m+1+\dfrac{b}{a}\right) \\ -\,\dfrac{k-j}{j+1} \, \ln a - \left(m+\dfrac{b}{a}\right) \, \ln(j+1)$ (unstable)

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

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \ln(an+b) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2} \,-\, \ln\!\Gamma\!\left(1+\dfrac{b}{a}\right)}_{\text{seed}} \\ \underbrace{-\,\dfrac{k-j}{j+1} \, \ln a \,-\, \dfrac{b}{a} \, \ln(j+1)}_{\text{spacing } \& \text{ shifting factor}}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \ln(n+b) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \ln\!\Gamma(1+b)}_{\text{seed}} \, \underbrace{-\,b\ln(j+1)}_{\substack{\text{spacing } \& \\ \text{shifting factor}}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \ln(an) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2}}_{\text{seed}} \underbrace{-\,\dfrac{k-j}{j+1}\,\ln a}_{\substack{\text{spacing } \& \\ \text{shifting factor}}}$ (unstable)

j-spaced monotonic divergent series of logarithm numbers

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \ln(an+b) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2} \,-\, \ln\!\Gamma\!\left(1+\dfrac{b}{a}\right)}_{\text{seed}} \\ \underbrace{+\,\dfrac{j}{j+1} \, \ln a \,-\, \dfrac{b}{a} \, \ln(j+1)}_{\text{spacing factor}}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \ln(n+b) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \ln\!\Gamma(1+b)}_{\text{seed}} \, \underbrace{-\,b\ln(j+1)}_{\text{spacing factor}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \ln(an) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2}}_{\text{seed}} \underbrace{+\,\dfrac{j}{j+1}\,\ln a}_{\text{spacing factor}}$ (unstable)

k-shifted monotonic divergent series of logarithm numbers

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \ln(an+b) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2} \,-\, \ln\!\Gamma\!\left(1+\dfrac{b}{a}\right)}_{\text{seed}} \underbrace{-\,k\ln a}_{\substack{\text{shifting} \\ \text{factor}}}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \ln(b+n) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \ln\!\Gamma(1+b)}_{\text{seed}}$ (semi-stable)

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \ln(an) = \underbrace{\dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2}}_{\text{seed}} \underbrace{-\,k\ln a}_{\substack{\text{shifting} \\ \text{factor}}} = \underbrace{\dfrac{1}{2} \, \ln\!\left(\dfrac{2\pi}{a}\right)}_{\text{seed}} \underbrace{-\,k\ln a}_{\substack{\text{shifting} \\ \text{factor}}}$ (unstable)

SECTIONS

$\displaystyle \sum_{n\geq1} \ln(an+b) = \dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2} \,-\, \ln\!\Gamma\!\left(1+\dfrac{b}{a}\right)$ (unstable)

$\displaystyle \sum_{n\geq1} \ln(n+b) = \dfrac{\ln(2\pi)}{2} \,-\, \ln\!\Gamma(1+b)$ (semi-stable)

$\displaystyle \sum_{n\geq1} \ln(an) = \dfrac{\ln(2\pi)}{2} \,-\, \dfrac{\ln a}{2} = \dfrac{1}{2} \, \ln\!\left(\dfrac{2\pi}{a}\right)$ (unstable)

INFRASECTIONS

$\displaystyle \sum_{n\geq m+1} \ln(an) = \sum_{n\geq 1} \ln(a(n+m)) = \underbrace{\dfrac{1}{2} \ln\!\left(\dfrac{2\pi}{a}\right)}_{\text{seed}} \,-\, \underbrace{\ln m!}_{\substack{\text{truncation} \\ \text{factor}}}$ (semi-stable)

$\displaystyle \sum_{n\geq m+1} \ln(an+b) = \sum_{n\geq 1} \ln(a(n+m)+b) \\ = \underbrace{\dfrac{1}{2} \ln\!\left(\dfrac{2\pi}{a}\right) \,-\, \ln\!\Gamma\!\left(1+\dfrac{b}{a}\right)}_{\text{seed}} \,-\, \underbrace{\ln\!\left(\dfrac{\Gamma\!\left(m+1+\dfrac{b}{a}\right)}{\Gamma\!\left(1+\dfrac{b}{a}\right)}\right)}_{\text{truncation factor}} \\ = \dfrac{1}{2} \ln\!\left(\dfrac{2\pi}{a}\right) \,-\, \ln\!\Gamma\!\left(m+1+\dfrac{b}{a}\right)$ (semi-stable)

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