A very simple question, but I can’t seem to find anything relating to it :
Is there any research, are there any results that have focused on or given insight on
A very basic series, converges extremely fast, its value is around .29. What more can there be said about it ?
From what little I know about more advanced number theory, similar sequences (I can think of a few similar ones that I can’t find any relevant research or results about) can be very non-trivial to compute or to analyse.
This is OEIS A094289, where they have no information except computations of the value. This suggests the answer “no” to the question “is there any research …”
What more can there be said about it ?
Essentially nothing. Related series are
Sophomore’s constant $\displaystyle C_s=\sum_{n=1}^{\infty}\frac{1}{n^n}$.
Prime zeta values $\displaystyle P(k)=\sum_{p\in\mathbb P}\frac{1}{p^k}$, with $k\in\mathbb N_{\ge 2}$.
No closed form expressions for these constants are known so far.
it may not be elegant but as an idea $\sum = \sum_{n=1}^{n=5}1/{p^p} + \sum_{n=6}1/{p^p}$
and as per this leverage approximation / boundaries
$\log{n} + \log{\log{n}} – 1 < \frac{p_n}{n} < \log{n} + \log{\log{n}}$ for $n \geq 6$
and then look at convergence of $\sum_{n=6}$