What are infinite series used for?
It is very tough to answer such a general question, but I will give it a shot. Infinite series allow us to add up infinitely many terms, so it is suitable for representing something that keeps on going forever; for example, a geometric series can be used to find a fraction equivalent to any given repeating decimal such as:
by splitting into individual decimals,
by rewriting into a form of geometric series,
by using the formula for the sum of geometric series,
The knowledge of geometric series helped us find the fraction fairly easily. I hope that this was helpful.