Formal power series

Imagine you have a very special type of calculator that can do math with an infinite number of terms. This calculator is called a formal power series.

Now, let's say you want to solve a mathematical problem that involves a function. A function is like a machine that takes in one number and gives you another number as output.

With a formal power series, you can represent a function as an infinite sum of terms. Each term has a power of x raised to some exponent, and a coefficient that multiplies that power of x.

For example, let's say you want to represent the function f(x) = 1 / (1 - x) as a formal power series. This means you want to write it as an infinite sum of terms that look like this:

a_n * x^n

Where a_n is a coefficient and n is a non-negative integer.

With some math magic, you can figure out that the coefficients for this particular function are:

a_n = 1 for all n

So the formal power series for f(x) is:

1 + x + x^2 + x^3 + ...

What this means is that you can use this infinite sum to represent the function f(x) for any value of x between -1 and 1. That's pretty cool!

Formal power series have a lot of applications in math and science. They can be used to solve differential equations, describe the behavior of physical systems, and more. But the basic idea is always the same: representing a function as an infinite sum of terms with powers of x and coefficients.