Suppose that $\{P_n\}$ are sequential functions of polynomials of degree $n$, and $x$ is between $[-1,1]$.
Put $P_0 = 0$, and define, for $n = 0, 1, 2, \ldots ,$
$$P_{n+1} = P_n + \frac{ (x^2 - P_n^2) }{2}$$
My question is: What is the the explicit formula of $P_n$?
I try this:
if $n=0$ then :
$$ P_1 = a_1 * x = \frac{ (x^2) } {2} $$
But this is wrong!! Because we know that $a_i$ is a constant real number like $0.5$! not function of $x$.
((My question is about chapter 7 of rudin analysis SEQUENCES AND SERIES OF FUNCTIONS question number 23)). It says that this polynomials convergent uniformly to |x|.
