# Example of sequence of functions

• May 1st 2011, 06:03 AM
token22
Example of sequence of functions
Hello,

Could you help me to solve http://i51.tinypic.com/2nhhiqp.png
Actually I am totally desperate of it and can't even start. I am not good in exaples such these ("give an example..."). It would be perfect if you type some steps how to find it http://www.mymathforum.com/images/smilies/icon_wink.gif
(I hope I translated it right)

Thanks a lot
• May 1st 2011, 07:11 AM
hatsoff
Hmm. I'm not sure if the interval <0,1> is supposed to denote (0,1) or [0,1]. If it's the former, just try $nx^{n-1}$. The latter is a bit trickier.
• May 1st 2011, 07:24 AM
token22
I think it should be [0,1]. Could you explain each possible sequence please.
• May 1st 2011, 07:28 AM
FernandoRevilla
For the latter, choose

$f_n:[0,1]\to \mathbb{R}\quad,\quad f_n(x)=nx(1-x^2)^n$
• May 1st 2011, 08:21 AM
token22
Thank (could you describe how you found it?). One more thing - I hope i can join it to this one thread. We have:

$\lim_{n \to \infty} \int_{0}^{\infty} { \frac{cos (\frac{x}{n})} {(1 + \frac {x}{n})^{n}} }$
I solved it as: $x\in [0,\infty)\:; { \frac{cos (\frac{x}{n})} {(1 + \frac {x}{n})^{n}} } \leq \frac{cos \, 0}{1^n} \:;\: x>0$ (Weierstrass test - uniform convergence) so i can change integral with limit and then it is easy - result = 1

Is this solution right ?
• May 1st 2011, 10:21 AM
FernandoRevilla
Quote:

Originally Posted by token22
Thank (could you describe how you found it?).

It is a well known counterexample for

$\displaystyle\lim_{n \to{+}\infty}{\int_a^b f_n}=\int_a^b \displaystyle\lim_{n \to{+}\infty}{f_n}$

when we consider pointwise convergence.
• May 1st 2011, 10:31 AM
FernandoRevilla
Quote:

Originally Posted by token22
Is this solution right ?

No, it isn't. Notice that for your bounds

$|f_n(x)|\leq M_n=1$

we have

$\displaystyle\sum_{n=1}^{+\infty}M_n\;\; \textrm{divergent}$
• May 1st 2011, 10:53 AM
token22
Then how can I prove that I can change integral with limit (it comes to me quite impossible to solve integral first anf then limit) ?
• May 1st 2011, 10:58 AM
TheEmptySet
Quote:

Originally Posted by token22
Then how can I prove that I can change integral with limit (it comes to me quite impossible to solve integral first anf then limit) ?

Do you know Lebesgue's dominated convergence theorem?

If so it will be very useful.
• May 1st 2011, 12:11 PM
token22
I can't find the function that is greater and its integral do not divergent. Maybe I can't change integral with limit.