Convergence of a Sequence of Functions

[felvt]

Construct an example of a pointwise limit of continuous functions that converges everywhere on the compact set [-5,5] to a limit function that is unbounded on this set.

Not sure where to start on this, but I have a picture in mind. I suppose I would need the functions to approach something similar to a parabola that goes to infinity as x goes to -5 and 5.

 

[Deleted member 4993]

Construct an example of a pointwise limit of continuous functions that converges everywhere on the compact set [-5,5] to a limit function that is unbounded on this set.

Not sure where to start on this, but I have a picture in mind. I suppose I would need the functions to approach something similar to a parabola that goes to infinity as x goes to -5 and 5.

Good thought....

\(\displaystyle G(x) \, = \, \frac{1}{(x+5)(x-5)}\)

should work

 

[felvt]

Right, but I am also having trouble coming up with the sequence of functions that would converge pointwise to a function like that.

One of the form Gn(x) with n=1,2,3,4......

 

[daon]

I'm not to sure about this... but instead of (x-5)(x+5) have (x-5(n-1)/n)(x+5(n+1)/n)

 

[Deleted member 4993]

How about using Taylor's series:

\(\displaystyle \frac{1}{1-x} \, = \, 1 \, + \, x \, + \, x^2 \, + \, x^3 ....\)