How to use Dirichlet/Abel test to prove the series is not uniform convergent

Question

Summation from n = 0 to infinity fn(x) where $$f_n(x) = \frac{x^2}{(1+x^2)^n}$$ and x is real is the series in question and I have to show it is convergent but not uniform convergent. I would appreciate help. Thank you.

Answer 1

The convergence is not uniformly convergent on a subset where $0$ is a limit point such as $(0,a)$.

Note that

$$\left|\sum_{k=n+1}^\infty\frac{x^2}{(1+x^2)^k} \right| \geqslant \sum_{k=n+1}^{2n}\frac{x^2}{(1+x^2)^k} \geqslant \frac{nx^2}{(1+x^2)^{2n}}, $$

and with $x = 1\sqrt{2n} \in (0,a)$ we have

$$\sup_{x \in (0,a)}\left|\sum_{k=n+1}^\infty\frac{x^2}{(1+x^2)^k} \right| \geqslant \frac{1}{2\left(1+\frac{1}{2n}\right)^{2n}}$$

Since the RHS converges to $1/(2e)\neq 0$ as $n \to \infty$, the Cauchy criterion for uniform convergence is violated.

Answer 2

What you give is a sequence. Check your definition, the sequence's uniform convergence depends on the interval of $x$ considered.