Are Taylor series always uniformly convergent? [closed]
Solution 1:
No, Taylor series are not always uniformly convergent. Consider, for example, the Taylor series to $$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots$$ This converges pointwise everywhere, but does not converge uniformly: No matter what $n$ is, $$e^x - \left(1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}\right)$$ is unbounded as $x \to \infty$.