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This year an exceptionally intellectual student took maths as a major subject for college. She contacted me for guidance on how to and what to study in mathematics . As she had only had a knowledge of high school mathematics , I asked her to wait till the more interesting and rigorous stuff is taught in her class. But she was really too interested mathematics so I decided to channelize her interest in a proper direction .I suggested her books in introductory number theory and the famous - $\text{What is Mathematics- Ian Stewart }$ . I also decided to give her a maths problem everyday and give up the solutions the next day before giving that day's problem . So here I would please need help :

What are some problems which are somewhat junior undergraduate level (in any branch) but expand our understanding , add tools in our mathematical toolbox and also teach us lessons which are later on useful as a professional mathematician.

Like I told her to tell me what $$\sum_{n=1}^{\infty}\frac{1}{n}$$ evaluate to ? And then revealed it is known as harmonic series and it diverges . To tell her about this problem , the fact there there is comparison test for convergence and the fact that what appears to be may not be true! Then I also gave her the famous differentiation under the integral sign to show that problem can be solved that way too . I have couple of them in mind .But it would be great help if I get more such problems .

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  • $\begingroup$ I'm now in my last year of high school and things like complex numers could be interesting, some things like juliasets or fourierseries could be interesting to look at. $\endgroup$ – stein Dec 7 '16 at 14:30
  • $\begingroup$ This set of problems could prove to be useful: wumath.wustl.edu/undergraduate/information-math-majors/… $\endgroup$ – Rohan Dec 7 '16 at 15:09
  • $\begingroup$ @Rohan it's great. But you see she should work it in a day $\endgroup$ – Shivam Patel Dec 7 '16 at 15:30
  • $\begingroup$ I'm confused about how you can simultaneously describe someone as a brilliant mathematics major in their junior year and also possessing a high school level of knowledge. Similarly, I'm shocked that she hasn't seen the divergence of the harmonic series before. $\endgroup$ – Stella Biderman Apr 4 '17 at 4:02
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These problems compiled by Prof. Laszlo Babai of the University of Chicago is excellent, in my opinion. See here for more sheets of puzzle problems.

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