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Examples of Uniformly, point wise convergence

Amer

Active member
Mar 1, 2012
275
I need some examples of sequences some converges uniformly and some point wise Thanks in advanced
 

chisigma

Well-known member
Feb 13, 2012
1,704
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
Jan 17, 2013
1,667
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 \(\displaystyle [0,b]\).