1. ## Series of function...

Let, $\displaystyle f_n(x)=xarctg(nx)$ when $\displaystyle 0\leq x<\infty$.

I need to find:
1. Limit function (Which I found: $\displaystyle \frac{\pi}{2}x$ )

2. Checking the unitary convergence. (So, here is my problem)

Thank you all!

2. Be careful! The limit function is not exactly $\displaystyle \frac{\pi}{2}x$. Make sure you consider two cases: $\displaystyle x\geq 0$ and $\displaystyle x< 0$.

Also, what do you mean by "unitary convergence"? Do you mean "uniform convergence"?

3. Yes, "uniform convergence".
We don't need x<0. x is always grater than 0 (or x=0)

Thank you!

4. Oops, sorry about that. I didn't see the domain restriction.

For uniform convergence, did you try looking at the definition? Given $\displaystyle \varepsilon>0$, is there an $\displaystyle N$ large enough so that $\displaystyle n>N$ implies $\displaystyle \left|x\arctan (nx)-\frac{\pi}{2}x\right|<\varepsilon$ for all $\displaystyle x\geq 0$?

5. Note that $\displaystyle \frac{d}{dx}(x.arctan(n x)) = \frac{nx}{n^2 x^2+1}+arctan(n x)$.
Look for the value of x for which the derivative of $\displaystyle |x.arctan(nx)-\frac{\pi}{2}x|$ (for x positive) is zero and draw the table of variations. Now, find the maximum value of that quantity over th given domain. You have uniform convergence if and only if the limit of the value you get is zero.

hope this helps