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1 hour ago comment added mathworker21 @SeanEberhard "The issue is they just state the result and do not give details."
9 hours ago comment added Sean Eberhard Sure, but the error $O(t^2)$ is at least good enough to check the sign you are interested in.
10 hours ago comment added Abdelmalek Abdesselam I upvoted but unfortunately this does not at all answer my question. I should have mentioned the references I looked at already which include the FS classic book, and will edit my question accordingly. The issue is they just state the result and do not give details. Also, they state the remainder as $O(t^2)$ which I find suspicious. If they really did the exercise, they should have seen the remarkable cancellations that make this remainder $O(t^{\infty})$.
11 hours ago history answered Sean Eberhard CC BY-SA 4.0