(adsbygoogle = window.adsbygoogle || []).push({}); Why does this "trick" work? Screenshot inside

I managed to do the computations but unfortunately I don't understand why the trick works in part b). It is said that N has to be very very big to find the answer within 0.005, yet in part b I find N ≈ 130 (which I know is right). That means the solution can now be found with error < 0.005 with only 130 terms. Is that right? Could someone explain to me what's going on brievely? This is a scan from Mary L Boas mathematical methods book which I'm self-learning from, and unfortunately this is the only information found on this trick to evaluate the error.

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# Why does this trick work? Screenshot inside

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