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## Homework Statement

I am trying to make sure I am using the Weierstrass test correctly. So, on the following expression: $$f_n(x) = \frac{x}{1+nx^2}, -1 \le x \le 1$$

I looked at it this way. First I checked to see what happens at -1 and 1. The "plug in" test gets me a limit of 1/2 at x=1 and n=1, and a limit of 1/2 at x=-1 and n=1. All-righty then, we do a little L'Hôpital's rule: $$\lim_{n \rightarrow \infty} f_n(x) = \frac{\frac{d}{dn}(x)}{\frac{d}{dn}(1+ nx^2)}= \frac {0}{x^2}=0 $$

So I know this converges to zero. But does it do so uniformly? Well, I can see that this function has a bound at x=1/2. So I will use [itex]\frac{1}{1+n}[/itex] as my M-function. Giving us:

[itex]M_k = \frac{1}{1+n},[/itex] and [itex]\sum_{n=1}^{\infty}\frac{1}{1+n}[/itex].

[itex]\frac{1}{1+n}[/itex] is always going to be smaller than [itex]\frac{x}{1+nx^2}[/itex] for any n with a particular x. We can check his out: since x is between -1 and 1, and n≥1, the [itex]1+nx^2[/itex] is always going to be less than 1+n.

So we can say $$ \left| \frac{x}{1+nx^2} \right| \ge \frac{1}{1+n}$$ for all n. In addition to that, [itex]\frac{1}{1+n} \lt \infty[/itex] for all n, but the other condition doesn't hold, so we don't have uniform convergence.

Did I get it right?

Thanks!