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 is bounded by 1 and the factor in front of it goes to zero, we get

Formally one could use for instance comparison.

Conclusion: The given sequence converges to the function f (x) = e^x/4 on the whole real line (which is thus the region of convergence of this sequence).

How about uniform convergence? We start by investigating the difference between f and a particular f_k on the above region of convergence.

We have just proved uniform convergence.

Conclusion: The given sequence converges to the function e^x/4 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.

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