From now on : Definition : [pmath size=18](u_{n})_{n in bbN}[/pmath] converges to [pmath size=18]l ~ doubleleftright ~ forall epsilon{gt}0, ~ exists p in bbN[/pmath] such that [pmath size=18]forall n{ge}p delim{|}{u_{n}-l}{|}{le}epsilon[/pmath]
Remark : in the case of a vector sequence, just replace [pmath size=18]delim{|}{u_{n}-l}{|}[/pmath] by [pmath size=18]delim{vert}{u_{n}-l}{vert}[/pmath] Taylor series with integral remainder : If [pmath size=18]f[/pmath] is a class, [pmath size=18]~ C^{n+1}[/pmath] is a class over the interval : [pmath size=18] f(x)=f(0)+xf{prime}(0)+{x^{2}}/{2!}f{prime prime}(0)+...+{x^{n}}/{n!}f^{n}(0)+int{0}{x}{{(x-t)^{n}}/{n!}f^{(n+1)}(t)dt}[/pmath] Fourier series : Definition : Then we call the Fourier series of [pmath size=18]f[/pmath] : [pmath size=18]S(f)(t)=a_{0}+sum{n=1}{+infty}{a_{n} cos(n omega t)+b_{n} sin(n omega t)}[/pmath] with [pmath size=18]omega={2pi}/{T}[/pmath]