If decreases to zero, then converges.
The alternating harmonic series converges. Its even partial sums increase, its odd partial sums decrease, and the two limits agree because their difference is the next term. In particular, every even partial sum is a lower bound and every odd partial sum is an upper bound for the limit.
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The Alternating Series Test is a method used to determine the convergence of an alternating series. An alternating series is a series of the form: \[ \sum_{n=1}^{\infty} (-1)^{n} a_n \] where \( a_n \) is a sequence of positive terms. The test provides a way to conclude that the series converges under specific conditions.