# Examples of Uniformly, point wise convergence

#### Amer

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

#### chisigma

##### Well-known member
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

$\chi$ $\sigma$

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
If $$\displaystyle f_n(x)$$ 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 $$\displaystyle f_n(x)=x^n$$ on the interval $$\displaystyle [0,1)$$ this sequence is not uniformally convergent but any closed subset is. Essentially we can use the M-test to prove uniform convergence. Choose $$\displaystyle [0,b] \subset [0,1)$$ then we have the following

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