Alternating series

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed $${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}$$ or $${\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}}$$ with a_n > 0 for all n.

Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a_n converge to 0 monotonically, but this condition is not necessary for convergence.