Transcription of Lecture Notes - Pennsylvania State University
{{id}} {{{paragraph}}}
Example: quiz answers
Lecture Notes - Pennsylvania State University
SPRING 2009 MATH 401 - NOTESS equences of functionsPointwise and Uniform ConvergencePreviously, we have studied sequences ofreal numbers. Now we discusssequences of real-valued functions. By a sequence{fn}of real-valued func-tions onD, we mean a sequence (f1, f2, .. , fn, ..) such that eachfnis afunction having domainDand range a subset Pointwise a subset ofRand let{fn}be a sequence of functionsdefined onD. We say that{fn}converges pointwise onDiflimn fn(x)exists for each other words, limn fn(x) must be a real number that depends only this case, we writef(x) = limn fn(x)for everyxinDandfis called thepointwise limit of the sequence{fn}.Formal Definition:The sequence{fn}converges pointwise tofonDif foreveryx Dand for every >0, there exists a natural numberN=N(x, )such that|fn(x) f(x)|< whenevern > :The notationN=N(x, ) means that the natural numberNdependson the choice ofxand.
Therefore, uniform convergence implies pointwise convergence. But the con-verse is false as we can see from the following counter-example. Example 10 Let {fn} be the sequence of functions on (0, ∞) defined by fn(x) = nx 1+n2x2. This sequence converges pointwise to zero. Indeed, (1 + n2x2) ∼ n2x2 as n gets larger and larger. So, lim n→∞ ...
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: