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The conditions for the integral test and comparison test state that in order to determine the convergence of a series, the terms need to be positive and decreasing 'eventually'. But how can a series have negative terms that become positive and then start decreasing and still be considered a series? Are there any examples of such a series?

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A series is just an infinite sequence of numbers that we sum up. Whether or not it converges or not does not matter.

For example, a convergent series that fits your requirements is

$$ \sum_{n=0}^{\infty}a_n $$ With $a_0 = -1, a_1 = 1, a_n = \frac{1}{n^2}\text{ for } n\ge 2$.

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