Solving Exercises in the book without a solution manual

Question

I'm currently a high school student, but I'm trying to study mathematics in an undergraduate(or graduate) course(especially analysis), because I'm interested on it. However, most of those books don't have a solution manual, and I have a difficulty to check if I solved the exercises correctly, and clearly. Is there a good way to check if my solutions are correct? If there's no way, then what things can I do instead of solving exercises in that book to improve myself on that subject? (Sorry for inappropriate tags. I couldn't think of the tags that are suitable to this question.)

Answer 1

This is a very common issue specialy among freshman and sophomore year students because, in general, math books intended to teach basic mathematics usually offer the answers (or at least part of it) at the end of the book but, as you pass to more advanced material, answers are no longer avaiable.

If you are in, say, your somophomore year, it's expected that you already got some intuition on whether your reasoning/resolution and answer is correct or at least make any sense or not. However, it might take some time for one to get such intuition and I completely understand if you don't have it now, since you are at the very begining of your math studies. Thus, let me give some tips on how you can help yourself.

(1) As I said before, many introductory books actualy have answers to their exercises so, if your book does not have them, you can probably find another one on the same topic which has.

(2) Math Stack is a very nice option to discuss and ask questions to other/more experienced people. You usualy will find very good help in here, and you can always post your questions here. It will probably help you a lot (it helps me a lot frequently).

(3) For some explicit calculation problems, you can check your answers at websites such as wolfram alpha. Say you want to solve an equation, a derivative, an intergral...you can always check your answers there.

(4) Discuss the questions and solution with your friends. This is the basic idea of this website, and it shows how valuable (and yet not so frequent done) this exercise is.

Well, this is all I could think of at the moment but surely other people can complement my answer here. Hope this helps!