- #1
twoflower
- 368
- 0
Hi,
next one I've little problems with:
[tex]
f_{n} = \frac{2nx}{1+n^{2}x^{2}}
[/tex]
[tex]
\mbox{a) } x \in [0, 1]
[/tex]
[tex]
\mbox{b) } x \in (1, \infty)
[/tex]
First the pointwise convergence:
[tex]
\lim_{n \rightarrow \infty} \frac{2nx}{1+n^{2}x^{2}} = 0
[/tex]
I computed the derivative of my function to know where to get supremum and I got that maximum is in [itex]x=\frac{1}{n}[/itex].
I used it to get supremum on the first interval and I got
[tex]
\sup \left\{\left|\frac{2nx}{1+n^{2}x^{2}}\right|, x \in [0,1]\right\} = 1 \Rightarrow f_{n} \mbox{ doesn't converge uniformly on } [0,1]
[/tex]
On the second interval [itex](1, \infty)[/itex] I cannot use the point [itex]x = \frac{1}{n}[/itex] and since the function is decreasing on this interval I will take the leftmost point to compute the supremum. It will be
[tex]
\sup \left\{\left|\frac{2nx}{1+n^{2}x^{2}}\right|, x \in (1,\infty)\right\} \leq \left|\frac{2n}{1+n^2}\right| \longrightarrow 0 \mbox{ as n goes to \infty}
[/tex]
Thus
[tex]
f_{n} \rightrightarrows 0 \mbox{ on (1, \infty)}
[/tex]
According to "official" results this is not complete result. Would you help me please to finish it?
Thank you.
next one I've little problems with:
[tex]
f_{n} = \frac{2nx}{1+n^{2}x^{2}}
[/tex]
[tex]
\mbox{a) } x \in [0, 1]
[/tex]
[tex]
\mbox{b) } x \in (1, \infty)
[/tex]
First the pointwise convergence:
[tex]
\lim_{n \rightarrow \infty} \frac{2nx}{1+n^{2}x^{2}} = 0
[/tex]
I computed the derivative of my function to know where to get supremum and I got that maximum is in [itex]x=\frac{1}{n}[/itex].
I used it to get supremum on the first interval and I got
[tex]
\sup \left\{\left|\frac{2nx}{1+n^{2}x^{2}}\right|, x \in [0,1]\right\} = 1 \Rightarrow f_{n} \mbox{ doesn't converge uniformly on } [0,1]
[/tex]
On the second interval [itex](1, \infty)[/itex] I cannot use the point [itex]x = \frac{1}{n}[/itex] and since the function is decreasing on this interval I will take the leftmost point to compute the supremum. It will be
[tex]
\sup \left\{\left|\frac{2nx}{1+n^{2}x^{2}}\right|, x \in (1,\infty)\right\} \leq \left|\frac{2n}{1+n^2}\right| \longrightarrow 0 \mbox{ as n goes to \infty}
[/tex]
Thus
[tex]
f_{n} \rightrightarrows 0 \mbox{ on (1, \infty)}
[/tex]
According to "official" results this is not complete result. Would you help me please to finish it?
Thank you.
Last edited: