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Let $a_n=\ln [1+(-1{)}^n/n]$ and $S_N={\sum }_{n=2}^N{a}_n$. Consecutive even and odd terms cancel exactly:ThusBoth parity subsequences have the same limit, so the full sequence of partial sums converges and the sum is zero. This check is necessary: cancellation in fixed pairs alone does not always prove convergence of the original series.

For absolute convergence, each pair instead contributesThe inequality follows from $\ln (1+u)={\int }_0^u(1+s{)}^{-1}ds\ge u/(1+u)$. The lower-bound series diverges by comparison with the harmonic series. Therefore the paired logarithmic series with zero sum has