Monotonic Divergent Series of Real Numbers

List of sums of divergent series of real numbers (where not all terms are rational numbers).

$\Gamma(\,)$ denotes the Gamma function.
$\zeta(z)$ denotes the Riemann zeta function.
$\zeta'(z)$ denotes the derivative of the Riemann zeta function.
$\zeta''(z)$ denotes the second derivative of the Riemann zeta function.
$\zeta^{(m)}(z)$ denotes the m-th derivative of the Riemann zeta function.
$\beta(\,)$ denotes the Dirichlet beta function.
$\psi_0(\,)$ denotes the digamma function.
$\psi_1(\,)$ denotes the trigamma function.

$e \approx 2.718281828459045235360287471352662497757247093699959574\ldots$ 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.
$C \approx 0.915965594177219015054603514932384110774149374281672\ldots$ denotes the Catalan constant.

$A_n \; (n \geq 0)$ denotes the Glaisher-Kinkelin constants (with $A_0 = \sqrt{2\pi} \;$ and $A_1 = A$ ).
$H_n$ 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.
$\gamma_n \; (n \geq 0)$ denotes the Stieltjes constants (with $\gamma_0 = \gamma$ ).

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