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## Homework Statement

I have already evaluated the integral 1/sqrt(x)exp(-ix) using The Residue Theorem and now I was looking for another method. So I thought of applying integration by parts and I got this attached formula.

Now I am wondering how to evaluate this series. My first doubt is whether it converges or not.

Actually I just want to evaluate it for n=1.

Does anyone have any ideas? Can this sum be expressed as an integral?

Thank you