Uniform convergence in series definitions of functions

Question

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!

Answer 1

"Tan"-ing time!

Consider on the interval $\left( -\frac{\pi}{2},\frac{\pi}{2} \right)$ the functions $f_n(x)$ defined for $n \in \Bbb{N}$ as $$ f_n(x) = \frac{2^{2n}(2^{2n}-1)B_{2n-1}x^{2n-1}}{(2n)!} $$ where $B_{2n-1}$ are the Bernoulli numbers. This is the series expansion of $$f(x) = \tan(x)$$ which is familiar enough.

For any given purported $N(\epsilon)$ to demonstrate uniform convergence, the sum of the terms up to $N(\epsilon)$ will be finite for any $x$ in this range, and in fact will be at most some finite value $M(N)$ which depends on $N(\epsilon)$. Then one can choose an $x$ sufficiently close to $\frac{\pi}{2}$ that $\tan x > M(N(\epsilon) + \epsilon$ which shows that $N(\epsilon)$ does not demonstrate uniform convergence.

So this series is not uniformly convergent on $\left( -\frac{\pi}{2},\frac{\pi}{2} \right)$.