No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms. However, it see

No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms. However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason. (Although, the higher the order of the zero, the more initial terms you'd need...) Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...