Divergent Series of Rational Numbers

List of sums of 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

$e \approx 2.71828182845904523536 \ldots$ denotes the Euler constant
and $\gamma = \zeta(1) \approx 0.57721566490153286 \ldots$ denotes the Euler-Mascheroni constant.