## Problems of functional series

1. Study the point convergence and the uniform convergence of the succession of functions with general term We first study the pointwise limit for . It must be taken into account when calculate the limit that the variable x can take any value in the domain of the function. Recalling that the function is bounded in the interval , and also for is bounded in the open interval whenever we get: Since when . In the following illustration we see the graphs of three terms: , in black, red and green, respectively, as the index increases the function tends to take smaller values:

Therefore is point-wise convergent to the null function . Let's now look at the uniform convergence.

Criterion of uniform convergence of a sequence  Converges uniformly to the point boundary whenever it is fulfilled: .

We apply it: . To find the maximum we derive and equate to zero, the absolute value does not concern us because the functions are symmetric respect to the abscissa axis (see previous figure): The first case is discarded because it provides a minimum not a maximum, since . For the second case, using the equality and calling to the point at which we have a maximum: We apply the limit: Therefore the sequence converges uniformly. 2. Study the pointwise convergence and the uniform convergence of the sequence of functions with general term Also study the uniform convergence in the constrained interval .

For small values of the fraction takes large values, and the , whereas for large values the opposite happens, . In the following image we have the graph of (in the point the image is not shown, which is zero): The transition point between the two values of can be easily found: ; At point we have , and we can then redefine the function definition: 