I was considering the sum $ \sum_{n=2}^{\infty} \frac{(-1)^n}{\ln(n)} $ which I realizes converges extremely slowly (due to the slow growing nature of the logarithm). I was curious if someone knew a closed form for this in terms of literally ANY OTHER functions? Plugging this into Wolfram alpha didn't reveal anything particularly interesting:
https://www.wolframalpha.com/input/?i=sum+%28-1%29%5En%2F%28ln%28n%29%29+n+%3D2+to+infinity
I suppose that a good place to start would be try to understand the function $$ \sum_{n=2}^{\infty} \frac{x^n}{\ln(n)} $$
And then evaluate it at $x=-1$. This function also was unfamiliar to me, so I tried to look at the even simpler object:
$$ \sum_{n=1}^{\infty} \ln(n)x^n $$
Which does happen to have a restatement as a lambert series: See here: https://en.wikipedia.org/wiki/Lambert_series#Examples (particularly the section called Von Mangoldt Function). But even that series seems to be dependent on properties of prime numbers (perhaps there might be a way to get the riemann zeta function involved).
Related:
Generally speaking $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^q} $ for small rational q (ex: $q = \frac{1}{2}$ or $q = \frac{1}{3}$) could be resolved using the riemann zeta function. The next slowest growing function that I could think of beyond polynomial roots was a logarithm which motivates this question.
Best Answer
Consider the series
$$f(x) = \sum_{n=2}^\infty \frac{(-1)^nn^x}{\log n}$$
Then
$$f'(x) = \sum_{n=1}^\infty (-1)^n n^x + 1 = 1 - \eta(-x) = 1 + (2^{1+x}-1)\zeta(-x)$$
where $\eta$ and $\zeta$ are the Dirchlet eta and Riemann zeta functions, respectively. Then the summation is given by
$$f(0) = \int_{-\infty}^0 1 - \eta(-x)\:dx = \int_0^\infty 1 - \eta(x)\:dx \approx0.9243$$