uniform convergence of series $\sum_{k=0}^{\infty} \frac{1}{1+k^2x} $

Question

I know that the series $\sum_{k=0}^{\infty}\frac{1}{\vphantom{\Large A}1\ +\ k^{2}\,x}$ is converges uniformly on $\left(a,\infty\right)$ for $a > 0$, but how can I show that it does not converge uniformly on $\left(0,\infty\right)$?

Answer 1

If the series is uniformly convergent on $(0,+\infty)$ then $$\lim_{x\to0} \sum_{k=0}^{\infty} \frac{1}{1+k^2x}= \sum_{k=0}^{\infty} 1\, \text{is finite}$$ which is obviously wrong. Conclude.

Added By the definition the series is uniformly convergent if $$0=\lim_{n\to\infty}\sup_{x\geq0}\sum_{k=n+1}^\infty \frac{1}{1+k^2x}\geq\lim_{n\to\infty}\sum_{k=n+1}^{2n}\frac{1}{1+k^2\times0}=\lim_{n\to\infty}n=\infty$$ which's wrong