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 .
