Aperiodic Monotonic Divergent Series of Rational Numbers

List of sums of aperiodic monotonic divergent series of rational numbers (where not all terms are integers).

$\Gamma(\,)$ denotes the Gamma function
$\zeta(\,)$ denotes the Riemann zeta function
$\psi^{(0)}(\,)$ denotes the digamma function

$H_n$ denotes the n-th harmonic number
$H_{n,r}$ denotes the n-th generalized harmonic number of order r
$H_n^{(r)}$ denotes the n-th hyperharmonic number of order r

$e \approx 2.718281828459045235360287471352662497757247093699959574\ldots$ denotes the Euler constant;
$\gamma \equiv \zeta(1) \approx 0.5772156649015328606065120900824024310\ldots$ denotes the Euler-Mascheroni constant;
$A \approx 1.2824271291006226368753425688697917277676889\ldots$ denotes the Glaisher-Kinkelin constant;
$C \approx 0.9159655941772190150546035149323841107741493742816721\ldots$ denotes the Catalan constant.

GENERA

$\displaystyle \sum_{n\geq0} \dfrac{1}{(n+a)^r} = \dfrac{(-1)^r}{(r-1)!} \, \psi^{(r-1)}(a) = \zeta(r) - H_{a-1}^{(r)} \qquad (a\in\mathbb{N^*})$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{(n+a)^r} = \dfrac{(-1)^r}{(r-1)!} \, \psi^{(r-1)}(a+1) = \zeta(r) - H_{a}^{(r)} \qquad (a\in\mathbb{N^*})$ (semi-stable)

Pages: 1 2 3 4 5 6 7 8