What kinds of things are infinite series useful for?

I'm a relative beginner studying Calc II, and I haven't been very motivated while learning infinite series.

Can you give some examples of how they're used in calculus or other areas of math, or in the sciences/"real world"? Well, either that, or a way of thinking about them that you personally find intuitive/fun (I know this second part is subjective, but I think the suggestions might still be valuable for me to try).

• I think by series being useful you mean power series, as they are the simplest things to work with. There are many types of series that are interesting but not useful for practical use, e.g. then common Arctan series that gives the value for $\pi$. Jan 30 '15 at 8:15
• If you do mean power series, then Taylor series are pretty excellent. Jan 30 '15 at 8:18

I think one of the most important ways in which series are useful are in approximating real-world phenomena. A simple example is the equation for a pendulum: $\ddot{\theta} = -\frac{g}{l} \sin \theta$. This ODE doesn't have a closed-form solution but by taking the Taylor series approximation of $\sin \theta \approx \theta$ we get a very standard, easy-to-solve differential equation $\ddot{\theta} = -\frac{g}{l} \theta$. This a "go-to" toy example in the world of physics, but series approximations are generally used when using the exact function or equation is too cumbersome and not necessary in terms of accuracy. If you can approximate a trigonometric function to just a few polynomials, and your bridge won't get anywhere close to collapsing, then why not?
For me, I have always been amazed at the relationship between infinite series and trigonometric functions. For example: $$\sin x = x - \frac{x^3}{3!}+\frac{x^5}{5!}+\cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}$$ and $$\cos x = 1 - \frac{x^2}{2!}+\frac{x^4}{4!}-\cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}$$ and many others. Of course, the tie-in to your question is that trigonometric functions have a very practical application in everyday life.