# $g_n(x)=\frac{\frac{x}{n\:}}{1+\frac{x^2}{n^2}\:}$ proving uniform convergence

$$g_n(x)=\frac{\frac{x}{n\:}}{1+\frac{x^2}{n^2}\:}$$

I have to show that it converges uniformly in $[0,\infty)$, I already got that it converges pointwise to $\frac{1}{2}$ but I am just stuck on uniformly proving it.

I have tried using Weierstrass M-Test and Abel's Uniform Convergence Test, but I always reach something not logical.

• You should revise your calculations regarding pointwise convergence. Note that for example, $g_n(0) = 0 \ (\forall n)$ so we do not have pointwise convergence to $\frac{1}{2}$. – qualcuno May 27 '18 at 12:04
• $g_n(\infty)=0$, so something is fishy here – tired May 27 '18 at 12:05
• To clarify . . . converges as $n\to\infty$ or as $x\to\infty$? – gen-z ready to perish May 27 '18 at 12:10
• I used the inequality of $a^2 + b^2 \ge 2ab$ and so it all got canceled neatly, can you pls explain why it is wrong? – David May 27 '18 at 12:32

## 1 Answer

$g_n$ does not converge uniformly. It converges pointwise to 0 and $g_n(n)=\frac 1 2$ so it does not converge uniformly.

• Thank for the answer, but now I have hit a wall in the last part of the question. we are given another function in the beginning of the question that is: $h_n(x) = \frac{xn}{1+x^2n^2}$ and it goes like this. define a new function $k_n(x) = g_n(x) * h_n(x)$, does it uniformly converge? I tried multiple algebraical ways to solve and simplify it but it just wont work. – David May 27 '18 at 18:16
• @David what does * stand for? (Convolution or just the product?). – Kavi Rama Murthy May 28 '18 at 5:36
• the product of both functions, probably should have left out the * – David May 28 '18 at 8:10
• @David You can easily see that $|h_n(x)| \leq \frac 1 {n^{2}}$ so $h_n \to 0$ uniformly. – Kavi Rama Murthy May 28 '18 at 9:00
• @David Sorry, there was a typo in the last comment. I meant $|k_n(x)| \leq \frac 1 {n^{2}}$. So $k_n \to 0$ uniformly. – Kavi Rama Murthy May 28 '18 at 9:50