What can you do with a Series?

[Mitchell]

I am currently taking Calc. II and up to this point in our text there has always been application problems associated with the Chapter sections. We are now in the section covering Series and sequences, I find the subject challenging, however, what can you do with them? What are the applications in real life? I would sure be gratefull if someone could answer this. My guess would be in the area of codes or some type of computer application?

[arildno]

Well, since a lot of functions can be written as an infinite series (a Fourier series representation, for example) a finite series approximation of that function (which might be the solution of some differential equation) might be on occasion the best approximation you've got.

Perturbation series solutions are common ways to derive an approximate solution of differential equations.

Asymptotic series approximations often converges extremely fast in their region of validity.

[Hurkyl]

Also, series are very easy to manipulate analytically. For example, they're trivial to differentiate and antidifferentiate. Their truncations are also trivial to integrate!