I am a tutor at university, and one of my students brought me this question, which I was unable to work out. It is from a past final exam in calculus II, so any response should be very basic in what machinery it uses, although it may be complicated. The series is: $$\sum \limits_{n=1}^{\infty} \frac{(-1)^n}{(2n+3)(3^n)}.$$
Normally I'm pretty good with infinite series. It is clear enough to me that this sum converges. None of the kind of obvious rearrangements yielded anything, and I couldn't come up with any smart tricks in the time we had. I put it into Wolfram and got a very striking answer indeed. Wolfram reports the value to be $\frac{1}{6}(16-3\sqrt{3} \pi)$. It does this using something it calls the "Lerch Transcendent" (link here about Lerch). After looking around, I think maybe I can understand how the summing is done, if you knew about this guy and special values it takes.
But how could I do it as a calculus II student, never having seen anything like this monstrosity before?