I am having some difficulties to understand the formal criterion for uniform convergence. In the book " Mathematics. It's content, methods and meaning" I stumbled with the following series
$$ x + (x^2 - x) + (x^3 - x^2) + ..., $$
The n-th partial sum is then
$$ S_n(x) = x^n $$
and therefore
$$S(x) = \lim_{x\to\infty} S_n(x) = \begin{cases} 0 & 0\leq x < 1 \\ 1 & x = 1 \end{cases}$$
Intuitively I understand how this series would diverge at x = 1 as it would become a sum of infinite ones. Then the criteria for convergence indicates
$$\eta_n = \sup_{0 \leq x \leq 1} |S_n(x) - S(x)| = 1$$
since $\eta_n$ does not convergence to 0 as $ n \to \infty$ the series is not uniformly convergent. Two doubts:
- Why is not enough knowing that S(x) does not converge to 0 for all the range $(0, 1)$
- How is possible $\eta_n = 1$, why is not 0? as
$$S(x) = \sup_{0 \leq x \leq 1} |S_n(x) - S(x)| = \begin{cases} |x^n - 0| & 0\leq x < 1 \\ |x^n - 1| & x = 1 \end{cases}$$
so as to $n \to \infty$ $x=1$ then $|x^n - 1| = 0$