I have to study
1 ) the simple convergence of
$$S(x) = \sum_{n=0}^{\infty} x^{2n}$$
and
2) the uniform convergence
My attempts :
1) $\forall x \in [0,1[$ $$S_n(x) = \frac{1-(x^2)^{n+1}}{1-x^2}$$
The series converges if and only if $|x| < 1$ so, $\forall x \in [0,1[$
$$S(x) = \sum_{n = 0}^{ \infty} x^{2n} = \frac{1}{1-x^2}$$
Can you check my solution and if there is an error can you help me?