The reason is: if converges , then (by a well known theorem) . So , converges .For the second one I would think that the sequence would converge because the sum diverges if the sequence diverges, so the sum cant converge if the sequence diverges. So True.
Choose the seriesFor the third one, I not sure at all about the convergence of Laurent series, where would I start with this one?
Yes, it is false. Another counterexample: and are entire functions and is not analytic at , so is not entire.And for the last, I believe its false because say if it was e^z/cosz then its not entire cause of the singularity at 0, but then i was thinking it would be entire so long as the function doesn't ever = 0 so I'm in a bit of a pickle. In the exam I would say false, because it doesn't specify any two entire functions.