0
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ 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 $\pi$. $\endgroup$ – Arjang Jan 30 '15 at 8:15
  • $\begingroup$ If you do mean power series, then Taylor series are pretty excellent. $\endgroup$ – Daniel W. Farlow Jan 30 '15 at 8:18
1
$\begingroup$

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?

On the complete opposite end of the spectrum, series are important for infinite-dimensional spaces of functions (like Hilbert spaces, which are fundamental to the study of quantum mechanics) in which the elements are expressed in terms of convergent infinite series. I think regardless of the area of math you stay in (as long as it's applied, numeric, or analytic in some fashion) series will play some role, somehow. Even if it's a minor role.

$\endgroup$
  • $\begingroup$ Thanks- both your and @induktio's answers were helpful, but I decided to choose yours over his because the end of your first paragraph led me to explicitly realize/re-realize that the partial sums of infinite series are really just ever-closer approximations of what they converge to, which ties into the beautiful idea of limits that calculus is founded on. $\endgroup$ – Asker Jan 30 '15 at 17:01
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ One should look at how useful are these series for practical computation. $\endgroup$ – Arjang Jan 30 '15 at 8:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.