Some Talks by Peter Jipsen 

MathCS Outreach » Talks » Wikis » Using Online Mathmatics Discussions in Moodle with the Discovery Method » Diagrams and graphs 
View  Edit  Links  History  Attachments 
A sequence of continuous functions that converges to a discontinuous function: (x,y) A sequence of differentiable functions that converge to a discontinuous function: (x,y) A sequence of functions for which the limit of the integrals is not equal to the integral of the limit: (x,y) And again a differentiable sequence with the same property (on the interval `[0,1]`): (x,y) `int_0^1n^2xe^(nx)dx=n^2xe^(nx)/(n)]_0^1int_0^1n^2e^(nx)/(n)dx`
`=n e^nn e^(nx)/n]_0^1=n e^ne^n+1=1(n+1)/e^n>1` and `n>oo`.
However, the pointwise limit of `n^2xe^(nx)` is zero, so the integral of this limit is not `1`.
(Interesting note: while creating this example I found an error in "Counterexamples in Analysis" where the sequence is given as `nxe^(nx)`. 
You are logged in as Peter Jipsen (Logout)