determining function series convergence

[johnk]

Hello,

I'm stuck on this problem:
Determine where does the function series f[sub:1zoubyv0]n[/sub:1zoubyv0] converge pointwise, uniformly and locally uniformly:
\(\displaystyle f_n(x) = n\cdot\left( \sqrt{x+\frac{1}{n}} - \sqrt{x}\right)\)
\(\displaystyle x \in (0, \infty)\)

Evaluating the limit
\(\displaystyle \lim_{n \to \infty} f_n(x)\)
I determined it converges to \(\displaystyle \frac{1}{2\sqrt{x}}\) pointwise.

Now I'm not sure how to determine uniform convergence. I tried finding \(\displaystyle \sup_{x\in (1,\infty)} abs(f_n(x) - f(x))\) (f(x) being the pointwise limit; to see if it goes to zero for n -> inf) by taking a derivative of that and finding for which x it equals zero, but that turned out to be very complicated, I'm not sure if it's the right way.

I'd appreciate help with the uniform convergence, and especially general pointers for this kind of excercises.

 

[royhaas]

Notice that your sequence looks suspiciously like the derivative of \(\displaystyle \sqrt{x}\). Multiply and divide by the "conjugate" and it simplifies the problem.

 

[johnk]

Okay, I got a bit scared of the complicated equation . It wasn't that bad.

I found out that \(\displaystyle x = \frac{1}{3n}\) is the maximum, when substituted into \(\displaystyle abs(f_n(x) - f(x))\), I got \(\displaystyle \frac{\sqrt{n}}{2\sqrt{3}}\).
\(\displaystyle \lim_{n\to\infty} \frac{\sqrt{n}}{2\sqrt{3}} \ne 0\), so it does not converge uniformly.

Now I'm not sure how to determine if it converges locally uniformly...

 

[royhaas]

I think "local uniform convergence" means you can find a small open set around a point such that you have uniform convergence in that open set. Does that help?

 

[johnk]

I think "local uniform convergence" means you can find a small open set around a point such that you have uniform convergence in that open set. Does that help?

I know the definition, but I don't know how to check if this series meets it. I guess I should somehow check points where uniform convergence fails and see if there is a neighbourhood where it converges uniformly?