Editing Picard–Lindelöf theorem (section)

== Example of Picard iteration ==
[[File:Picard iteration example tan x.svg|thumb|Four Picard iteration steps and their limit]]
Let <math>y(t)=\tan(t),</math> the solution to the equation <math>y'(t)=1+y(t)^2</math> with initial condition <math>y(t_0)=y_0=0,t_0=0.</math> Starting with <math>\varphi_0(t)=0,</math> we iterate 

:<math>\varphi_{k+1}(t)=\int_0^t (1+(\varphi_k(s))^2)\,ds</math> 

so that <math> \varphi_n(t) \to y(t)</math>:

:<math>\varphi_1(t)=\int_0^t (1+0^2)\,ds = t</math>

:<math>\varphi_2(t)=\int_0^t (1+s^2)\,ds = t + \frac{t^3}{3}</math>

:<math>\varphi_3(t)=\int_0^t \left(1+\left(s + \frac{s^3}{3}\right)^2\right)\,ds = t + \frac{t^3}{3} + \frac{2t^5}{15} + \frac{t^7}{63}</math>

and so on. Evidently, the functions are computing the [[Taylor series]] expansion of our known solution <math>y=\tan(t).</math> Since <math>\tan</math> has poles at <math>\pm\tfrac{\pi}{2},</math> it is not Lipschitz continuous in the neighborhood of those points, and the iteration converges toward a local solution for <math>|t|<\tfrac{\pi}{ 2}</math> only that is not valid over all of <math>\R</math>.