Calculate Picard iteration and prove converge to $e{^{x^2}}-1$

Question

Calculate picard iteration and prove converge to $e{^{x^2}}-1$ in the next problem:

$y'=2x(y+1)$, $y(0)=0$.

My work: Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ defined by $f(x,y)=2x(y+1)$

Then

$y_0=0$
$y_1(x)=0+\int_0^x2sds=x^2$
$y_2(x)=0+\int_0^x2s(s^2+1)ds=\frac{x^4}{2}+x^2$
$y_3(x)=0+\int_0^x2s(\frac{s^4}{2}+s^2+1)ds=\frac{x^6}{6}+\frac{x^4}{2}+x^2$

I can't find $y_n(x)$ can someone help me?

Answer 1

Write the last iterate you found as $$y_3(x)=\frac{x^{2⋅3}}{3!}+\frac{x^{2⋅2}}{2!}+\frac{x^{2⋅1}}{1!},$$ this should strongly hint to what the general form of $y_n$ looks like.