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we can leave out the first two terms? how come we can do that? is it because the series goes to infinity? can we do that for all taylor series even if we don't need to do it?

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If you want to test convergence of any series, only the "tail" matters, and you can leave out as many terms as you want (as long as it is a finite number of terms, of course).

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The idea is that the series must be smaller than the series without its first two terms. So if the series without its first two terms is a finite number, anything smaller than it must also be finite because if it were not, then the series without its first two terms would be infinite meaning that there is a contradiction somewhere. This is when you are showing that a series is convergent or divergent.

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