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!
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!
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 $\displaystyle x\geq 0$?
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