<s>[INST] What are some practical uses of power series? Why do we care about power series? Why are they important?
 [/INST] Possibly the most basic reason why power series are useful is that they allow you to approximate any (possibly convoluted, but smooth) function $f(x)$ by its (truncated) power series $f(x) = \sum_n a_n x^n = \sum_{n \leq N} a_n x^n + O(x^{n+1})$. Power series are easy to manipulate and understand, general functions - not necessarily.
Say you want to understand the behaviour of $f(x) = \frac{x \sin x}{ 1 - \cos x}$ near $x = 0$. Just expand $\sin x = x - \frac{x^3}{6} + O(x^5)$, $\cos x = 1 - \frac{x^2}{2} +  \frac{x^4}{24} + O(x^6)$ and find that
$$f(x) = \frac{x^2 - \frac{x^4}{6} + O(x^6)}{\frac{x^2}{2} - \frac{x^4}{24} + O(x^6)}
= \frac{1- \frac{x^2}{6} + O(x^4)}{\frac{1}{2} - \frac{x^2}{24} + O(x^4)}
\\ = (1- \frac{x^2}{6} + O(x^4))(2 - \frac{x^2}{6} + O(x^4))
= 2 - \frac{x^2}{2} + O(x^4)
$$
So, with little computational effort, the complicated function that we start with turns out to behave "just like" the simple polynomial $2 - \frac{x^2}{2}$. In particular, it has limit at $0$ equal to $\lim_{x\to 0} f(x) = 2$, and is concave. If you wanted to say how much is $\int_0^t f(x) dx$, you could just integrate  $2 - \frac{x^2}{2}$, and the error would not be larger than $O(t^5)$. If you wanted to compare this to, say, $g(x) = 2 \cos x = 2 - x^2 + \dots$, then just looking at the coefficients you see that $f$ is greater, for $x$ sufficiently small. 
Of course, the same can be redone for higher degrees of accuracy if one needs smaller errors.
</s>