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This is a general question, but infinite series is one of my favorite topics in calculus, and I've been reading posts on unknown convergent/divergent series, and many of them seem to include trig functions and prime numbers. Are there types of "elementary" (elementary meaning functions taught in most calculus courses) infinite series (that don't involve the above mentioned classes of functions in the summands, and besides the zeta function) whose convergence or divergence we don't know yet, including partial progress on those types of sums?

user86111
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  • I don't have a link, but I think to remember that the convergence of the series of reciprocals of twin primes depends on some other conjecture, or something like that. Maybe someone else can add details, if they know? – Angelo Rendina May 10 '15 at 19:54
  • Roughly speaking, we can determine convergence or divergence for any series whose summand is made out of functions that are monotonic and have monotonic derivatives. Functions that oscillate, like trig functions and functions involving (gaps between) prime numbers, can cause trouble, especially when the summand is set up to magnify the effect of the oscillations (so $\sum \frac{\sin n}{n^2}$ is no problem, but $\sum \frac1{n^2\sin n}$ is more difficult). – Greg Martin May 10 '15 at 19:55
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    And possible duplicate of http://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown?rq=1 – Angelo Rendina May 10 '15 at 19:55
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    The series of reciprocals of twin primes does converge - this is Brun's theorem. – Greg Martin May 10 '15 at 19:55
  • Possible duplicate of Are there any series whose convergence is unknown? – Klangen Dec 17 '18 at 11:31

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