To prove $\sum_{n=0}^{\infty}x^n$ does not unif. converge on $(-1,1)$, let $f_n$ be the partial sum, can I use the difference $|\frac{1}{1-x} - f_n|$ diverges?
2026-04-07 21:37:54.1775597874
Prove $\sum_{n=0}^{\infty}x^n$ not converge uniformly
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We have $|\frac{1}{1-x}-\sum_{n=0}^N x^n|= \frac{|x|^{N+1}}{1-x} \to \infty$ for $x \to 1$. This shows that $(\sum_{n=0}^N x^n)_N$ does not converge uniformly to $\frac{1}{1-x}$ on $(-1,1)$.