Monotonic Divergent Series of eta Function

SPECIES

$\displaystyle \sum_{n\geq0} (\eta(n)-1) = -\dfrac{1}{2}-\ln2 \qquad \textit{(convergent)}$

$\displaystyle \sum_{n\geq0} \eta(n) = -1 - \ln2 \qquad \textit{(unstable)}$

SUBSPECIES

$\displaystyle \sum_{n\geq1} (\eta(n)-1) = -\ln2 \qquad \textit{(convergent)}$

$\displaystyle \sum_{n\geq2} (\eta(n)-1) = 1 - 2\ln2 \qquad \textit{(convergent)}$

$\displaystyle \sum_{n\geq1} \eta(n) = -\dfrac{1}{2} - \ln2 \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq2} \eta(n) = \dfrac{1}{2} - 2\ln2 \qquad \textit{(unstable)}$

Pages: 1 2 3 4