Thanksgiving break update

Today Mohanji and I covered more stuff on Analysis. We covered the Cauchy Criteria for both series and sequences of functions. It almost resembles the one for sequences. One good thing about uniform convergence is -
If there are sequences of functions converging uniformly the resulting function is continuous.
Similarly
If there are sequences of partial sums of functions converging uniformly the resulting function is continuous. I will post the maple output of both shortly. The example for sequence of functions converging uniformly is 1/(n(x^2+1)). The sequence of functions converge to f(x) = 0.
The example for partial series of sequence of functions converging uniformly can be given by a slight modification of the above function 1/(n^2(x^2+1)). Note that this function has the highest value at x =0 and when n = 1. So its bounded. A discussion with Gayan revealed that the sum of the series 1/n^2 is Pi^2/6. We were not sure how to prove it that it well be bell shaped. But definitely its not difficult to prove that its going to be uniformly convergent.
Yesterday's fruitful discussion with Mohanji had me cleared that if sequence of functions converge uniformly to a function than that function is continuous and when its uniformly convergent its easy to show that epsilon/3 style proof.
Discussion with Kathleen over 319 was good. Now I have all the study problems marked and we were able to solve couple of them. So the goal for tomorrow should be to do more of 319 and wrap up with the proof of Wierstrass M test. Which makes use of comparison test. That if every term of a sequence is less than corresponding term of another sequence and if that other sequence converges than the former sequence converge. The comparison is easy to proof. It can be proved in two ways. One using Cauchy convergence and other using the Sigma notation and making use of bounded monotone sequence.