Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians’ attempts to answer them.

Calculusis not known to be such a field, as far as I know. (For now, let’s just assume this means the basic topics included in the staid and stagnant conventional first-year calculus course.)What are

- the most prominent and
- the most readily comprehensible
questions that can be understood by those who know the concepts taught in first-year calculus and whose solutions are unknown?

I’m not looking for problems that people who know only first-year calculus can solve, but only for questions that they can understand. It would be acceptable to include questions that can be understood only in a somewhat less than logically rigorous way by students at that level.

**Answer**

1) Convergence of the Flint Hills series

∞∑n=11n3sin2n

is unknown. One can also ask the same question with different exponents – see this paper for more details.

2) Closely related (although lim inf is typically not covered in first year calculus courses, it’s too not much of a stretch): whether or not

\liminf_{n \to \infty} |n \sin n| = 0

**Attribution***Source : Link , Question Author : Michael Hardy , Answer Author : zcn*