This looks like the sort of equality Ramanujan came up with.

Srinivasa Ramanujan - Wikipedia, the free encyclopedia

Your example is called an infinite series. They play an important part in calculus.

At the basic level number theory is, loosely, the study of integers, primes and arithmetic. Do you remember divisibility criteria of certain integers? For example: if the sum of the digits of a number is divisible by three, then that number is divisible by 3. These sorts of rules, and how to prove them, are what you might learn the first week of a number theory course. Things get more complicated.

As number theory advanced, tools from other disciplines were brought in to answer questions. One branch of number theory, analytic number theory, uses the theory of calculus (called analysis) and complex numbers (remember i?), to work on problems in number theory. I think your example is the result of work in analytic number theory.

Unfortunately, it can be very difficult to understand what is going on in difficult number theory. Your course in calculus most likely won't venture in that direction, as it requires a lot of additional background.

On the upside, there are a lot of interesting and beautiful theorems that you will learn on the way to analytic number theory, should you choose to major in mathematics in college. You can even start learning basic number theory now!

There are a lot of good books which give a gentle introduction to number theory. Perhaps you can ask one of your math teachers if they can give you some help while you work through one. Or perhaps your parents can find you a tutor. There are some very smart number theorists on this message board, and I'm sure everyone would be happy to help you if you got stuck on a problem or concept.