Thread: Sequence of functions convergence.

Alright I am having some trouble with this topic.

Let:

x belonging to ]1,+infinity[

I have to study pointwise and uniform convergence.
So I'd like to start with the pointwise and here it is:



So

Now I have to check the uniform convergence, so I'd begin this way:



Okay, here is where I am stuck, what do I have to do now? Do I have to take the first derivative respect to x and make it equal to zero? I am very confused. Thank you!

Your computation of pointwise convergence is correct: for any given , you have

As to the uniform convergence, you need to check the limit

What is the supremum here? Clearly, for values of approaching from above, the function is unbounded. In fact,

Therefore,

from which we conclude that the convergence is not uniform (but only pointwise).

What if we get a restriction to an interval such as [a,+infinity[ with a > 1 , is the convergence uniform in that case?

Yes. In such case, the supremum (maximum) would be attained at because is positive and monotonically decreasing in :

And this expression tends to zero for .

Alright thank you, I understand. So could you (if I am not asking too much) tell me what are the "usual" steps to take ? When do I have to take the first derivative? Thanks

I wouldn't know of any "usual steps". The procedure is roughly the same every time you need to check uniform convergence: look out for singularities or discontinuities (emerging in the limit ) and check if they lead to a failure of uniform convergence. I am not sure what you mean by "first derivative". There is no differentiation involved in what we have discussed above.

I recommend you keep in mind the two typical textbook examples where uniform convergence fails:





In a similar way, both series fail to be uniformly convergent. That is because, although we are eventually locked down to zero for every (pointwise convergence), near there always remains some little jump up to the value 1, which is such that the supremum of is always and will never go down to zero.

Note that the former function is continuous (like a bezel, or a ramp on the floor), while the latter function is discontinuous (like a step). So in the former case, the failure of uniform convergence goes along with an ever-increasing slope (the "first derivative" you may have meant), while in the latter case, it goes along with a discontinuity which is not resorbed for .