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I suppose the "real" point of the question was to show how the integral test works. Which is important, but doing so with equations where it's not necessary, especially giant messes like the recommended solution seem like a waste of time.
It's actually worse than nothing, because it encourages operational reasoning rather than deduction. Maybe okay for physicists, but not math / C.S. majors.
(Apologies to physicists, you guys are the best at manipulating series )