Problem: Investigate convergence of the following sequence of functions:
Solution: First we investigate pointwise convergence. We treat x as a parameter and evaluate limit with respect to k. Since the sine in the numerator is bounded by 1 in absolute value and the denominator goes to infinity, we get
Formally one could use for instance comparison.
Conclusion: The given sequence converges to the function
How about uniform convergence? We start by investigating the difference between f and a particular fk on the above region of convergence.
We have just proved uniform convergence.
Conclusion: The given sequence converges to the function 0 uniformly on the whole real line.
I asked the computer for a few graphs just to show the general idea of what is happening here.
This example is so simple that you probably wonder why are we showing this; there surely is something more to this problem. And there is, which we quickly find once we try to differentiate.
With exception of point
To sum up, we have functions fk that converge uniformly to f on any subinterval of the positive half-axis, but fk′ do not converge to f ′ on any such interval, not even pointwise. This shows that differentiating convergent series may be really tricky.