Problem: Investigate convergence of the following series of functions:

Solution: First we investigate pointwise convergence. We treat x as a parameter and test convergence of the resulting series of real numbers. In a typical case we would test absolute convergence using some tests, we show this approach in this note. However, this approach, while more general, yields less information, in particular it does not tell us what a sum of a series is. Usually we accept this limitation and go to tests right away, why not here?

If they did not expect us to do something with the sum of this series, just work out convergence, then they would have given this series without indexing limits, since they are irrelevant when we just check on convergence. The fact that those limits are there is a clear hint. We therefore need a better tool and we can get one with a smart trick. If we denote the fraction inside the power as y, we get a geometric series. We know when a geometric series converges, it is exactly when | y| < 1; then this convergence is also absolute. Therefore the gives series converges (absolutely) exactly if

Using the geometric series we can also find out what the sum of this series is.

Conclusion: The given series converges on the region of convergence (−1/2,∞) to the function f (x) = 1 + x. The convergence it absolute there, so (−1/2,∞) is also the region of absolute convergence.

How about uniform convergence? We start by investigating the difference between f and a certain partial sum s_N on the above region of convergence. Again, we use the fact that this difference is known for geometric series and the substitution y = x/(1 + x).

Since the supremum does not go to zero for N going to infinity, we do not have uniform convergence. Where does the trouble happen? If we investigate monotonicity of the function in the second last supremum, we see that it is increasing on (−1,1) if N is even; or it is decreasing on (−1,0] and increasing on [0,1) if N is odd. In both cases the value at 0 is zero. When we put it in absolute value, we see that the expression in the last supremum starts with the value 1/2 at (−1)⁺ and decreases down to zero (which happens at the origin), then it starts increasing and goes to infinity at 1^-. Therefore, if we want to force the supremum to go to zero for N growing, we have to cut off these two ends. In terms of x it means that we have to cut away the endpoints of the region of convergence −1/2 and infinity. Does it help? Fix any real numbers a, b such that −1/2 < a < b and consider the set M = [a,b]. How does this set translate to the language of y? Note that the function x/(1 + x) is increasing on the region of convergence, so the set M translates to the interval [a/(1 + a),b/(1 + b)] for y; notice also that both endpoints in this interval are less than 1 in absolute value; this will be important when it comes to a limit. Now we are ready to estimate.

Conclusion: The given series converges to the function f uniformly on sets of the form M=[a,b] for any a, b satisfying −1/2 < a < b.

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