2
$\begingroup$

If a function has infinite order derivative at $0$ and $\lim_{n\to \infty}(f(x)-\sum_{i=1}^{n} a_{n}x^n)=0$ for every $x \in (-r,r)$,then it can be expand as power series$\sum a_{n}x^n$,

My question is if this function is not differential at $0$,how to expand it as $\sum a_{n}x^n$ satisfied with $\lim_{n\to \infty}(f(x)-\sum_{i=1}^{n} a_{n}x^n)=0$ for every $x \in (-r,r)$?Is it unique ?

  • 0
    If one of the derivatives is infinite, then how do you obtain $a_n$, seeing that it depends on the derivatives themselves...2012-04-15
  • 0
    Re-derive Fourier series?2012-04-15
  • 1
    Check out the basics [linked here](http://en.wikipedia.org/wiki/Fourier_series). Your question was something that spurred decades of math research and resulted in radically changing the notion of a function. Fourier series is one method to get some kinds of convergence properties for series expansions of non-differentiable functions, when the points causing non-differentiability are sufficiently well-behaved.2012-04-15

2 Answers 2