Power series are very useful tools in calculus that help us understand different mathematical problems.

A power series looks like this:

$$\sum_{n=0}^{\infty} a_n (x - c)^n,$$

Here, $a_n$ are special numbers called coefficients, and $c$ is the center of the series.

You can use a power series to get a good estimate of a function within a certain range, called the interval of convergence. This range is decided by something known as the radius of convergence, or $R$. This radius tells us the values of $x$ where the series works well.

In simpler terms, power series let us write functions like $e^x$, $\sin(x)$, and $\ln(1 + x)$ in easier forms. For example, if we take the Taylor series expansion for $e^x$ around 0, we get:

$$\sum_{n=0}^{\infty} \frac{x^n}{n!}.$$

This series works for all real numbers $x$, making it a great way to approximate the function.

We can do several things with power series. For instance, we can add, subtract, or multiply them. We can also differentiate (find the rate of change) or integrate (find the area under the curve) each term, as long as we stay within the interval of convergence.

So, if you have two functions shown as power series, you can find their sum and product just by combining their series.

Being able to work with power series gives us more tools to solve calculus problems and helps us see how functions behave around their centers.