/sci/ - Science & Math

[Anonymous No.3831561]

I need help from someone who is familiar with convergence proofs. I think there is a mistake in the question, but don't know for sure.

Task is:
for n element of N (naturals) let f_n : [0,1] -> R (reals) be the function f_n(t) = t^n. The task is to show that the sequence
(f_n) converges to 0. The way I see it, it cannot converge (to 0) if t = 1, because all elements of (f_n) would be 1. Is there a
mistake in the question i.e. it should be f_n : [0,1) -> R or am I not getting something? Although this is homework help,
I am not asking you to do it for me, just tell me if the question should be [0,1).

[Anonymous No.3831580]

Shameless selfbump

[Anonymous No.3831597]

(f_n) is a sequence of function so it would not converge to the number 0 but instead to the function f : [0,1] -> R where f(1)=1 and f(t)=0 for t=/=1.

[Anonymous No.3831616]

>>3831597
Well, you can see the question here: (http://mathsnotes.math.ntnu.no/mathsnotes/show/homework+2011+6)), and it clearly says converges to 0. So your opinion is that there is a mistake in the formulation of the question?

[TN5 !/sci/TN5.. No.3831645]

What the hell is that? Sequences of functions don't just "converge", there are several definitions of convergence that lead to different results. The most common are pointwise and uniform convergences.

However, pointwise convergence is weaker, and your sequence doesn't even converge to 0 pointwise (because of t=1, as you've noticed). So it doesn't converge to 0 uniformly either.

I don't remember all the definitions of convergence, but here, for that convergence to happen, you need a definition in terms of the integral of the absolute value of the difference. I think that exists, I don't remember the name though. It would be something like, f_n converges to f on D iff
<span class="math">\int_D|f_n-f|\to 0[/spoiler].

[Anonymous No.3831647]

>>3831616
Ah, yes. By using this inner product you get that (f_n) will converge to the 0-function (ie f(t)=0 for all t) in the corresponding norm.

[TN5 !/sci/TN5.. No.3831660]

>>3831647
Oh, I didn't notice the definition of the inner product and only read the question. Ignore >>3831645 then, the convergence is properly defined in the homework, just not in OP's post. And indeed, there is no mistake.

I guessed the convergence's definition though.

[Anonymous No.3831672]

Haha. Typisk sivinger å ikke skjønne dette. Hold deg til brukerkursene dine du.

[Anonymous No.3831679]

>>3831660
>>3831647
Thanks, guys!