Conditional convergence

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

The terms at $n=2j$ and $n=2j+1$ are $\ln [(2j+1)/(2j)]$ and its negative. Thus the partial sums ending at an odd index are zero, while those ending at $2m$ equal $\ln [(2m+1)/(2m)]\to 0$. Considering both subsequences proves convergence of the full sequence of partial sums, not merely of grouped terms.

The paired sum of absolute values is $2\ln [1+1/(2j)]$. Since $\ln (1+u)\ge u/(1+u)$ for $u\ge 0$, these pairs dominate $2/(2j+1)$, whose series diverges by comparison with the harmonic series. The original series therefore has conditional convergence with the displayed sum.

A conditionally convergent real series can be rearranged to converge to any prescribed real value or to diverge. Its positive and negative subseries have infinite total magnitudes: successively add positive terms to cross a desired value and negative terms to cross back. Since the terms tend to zero, the overshoots shrink to zero.

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.

The Riemann series theorem specifies the possible sums of rearrangements of a conditionally convergent real series.

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