MHB Examples of Uniformly, point wise convergence

[Amer]

I need some examples of sequences some converges uniformly and some point wise Thanks in advanced

 

[chisigma]

I need some examples of sequences some converges uniformly and some point wise Thanks in advanced

A very suggestive example is given by the sequence of functions...

$\displaystyle s_{n} (x) = \sum_{k=0}^{n} (1-x)\ x^{k}\ (1)$

For $0 \le x < 1$ is $\displaystyle \lim_{n \rightarrow \infty} s_{n} (x) = 1$ but for x=1 is $\displaystyle \lim_{n \rightarrow \infty} s_{n} (x) = 0$, so that $s_{n} (x)$ conveges pointwise in [0,1) but doesn't uniformly converge in [0,1)...

Kind regards

$\chi$ $\sigma$

 

[alyafey22]

If $$f_n(x)$$ is uniformally convergent then it is point-wise convergent. The difference is that uniform convergence is defined on sets.

Take for example the sequence of functions $$f_n(x)=x^n$$ on the interval $$[0,1)$$ this sequence is not uniformally convergent but any closed subset is. Essentially we can use the M-test to prove uniform convergence. Choose $$[0,b] \subset [0,1)$$ then we have the following

$$x^n \leq b^n \,\,\, \forall \,\, x \in [0,b]$$ since $$\lim b^n = 0 $$ .$$f_n $$ is uniformally convergent on $$[0,b]$$.