Hints for Pledged assignment #2:

 

Exercise 1: Well, three or four of these come from examples that exist in the notes, text, or previous assignments. Two or three require some new thinking.

 

Exercise 3: This would be easy if the problem asked you to show that 1/f_n converges pointwise to 1/f. You could just say "Algebraic Limit Theorem" and be done. So the focus here is on showing uniform convergence. Given e>0, what do you want to make less than e? And what do you know you can make small?

 

Exercise 4: This particular problem grows naturally out of something you have done in a previous problem set.

 

Exercise 5: (a) Existing theorems we have proved about power series tell us most of what we want to know. For instance, we know that power series converge uniformly on closed intervals contained in their intervals of convergence. We also know that if the differentiated series converges uniformly, then the original series was differentiable and the derivative is in fact what we get when we differentiate term by term. So, all that's left to show is that taking the term by term derivative of a series yields a new series that converges pointwise on (-R,R). These steps are set out for you in exercise 6.5.7 in the text.

As some additional advice for part (a), take a few minutes to really understand the proof in the text for Theorem 6.5.1.