Conditional convergence

A series is conditionally convergent when it converges but does not converge absolutely.

If $a_n\ge 0$ decreases to zero, then ${\sum }_{n=1}^{\infty }(-1{)}^n{a}_n$ converges.

The alternating harmonic series ${\sum }_{n\ge 1}(-1{)}^{n+1}/n$ 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.

A rearrangement changes the order of a series without changing its multiset of terms. A conditionally convergent real series can be rearranged to approach any prescribed real limit or to diverge.