KnowraAlternating harmonic seriesAlternating harmonic seriesThe series \(1-\frac12+\frac13-\frac14+\cdots\), whose terms alternate in sign and whose sum is \(\ln 2\).BriefConnectAlternating series test: A convergence test for alternating series whose term magnitudes decrease to zero. The reciprocal terms decrease to zero, so this test proves convergence.Conditional convergence: Convergence of a series whose series of absolute values diverges. The series converges, but its absolute-value series is the divergent harmonic series.Infinite series: A sum defined as the limit of the partial sums of a sequence of terms. The alternating harmonic series is evaluated through its partial-sum limit.Mercator series: The power series expansion of the natural logarithm around one. Its endpoint evaluation yields this series and the value \(\ln 2\).Leibniz criterion: A theorem stating that an alternating series converges when its term magnitudes decrease to zero. This criterion applies directly to the series’ reciprocal terms.Absolute convergence: Convergence of a series of terms’ absolute values, which guarantees convergence under rearrangement. Taking absolute values destroys convergence in this example.Harmonic numbers: The numbers \(H_n=1+\frac12+\cdots+\frac1n\). Its partial sums equal \(H_{2m}-H_m\) after an even number of terms.Numerical analysis: The study of algorithms for approximating mathematical quantities with finite computation. Truncating the series gives a simple approximation with a rigorous error bound.Power series: An infinite series whose terms are powers of a variable multiplied by coefficients. The series is obtained by evaluating the logarithm’s power series at one endpoint.Riemann rearrangement theorem: A theorem that conditionally convergent real series can be rearranged to converge to any real number or diverge. Reordering these terms can produce different sums or divergence.Show all 21Linked from 9 pagesAbel's theorem (power series)Broader topic: Its power series gives a classic limit argument for evaluating the endpoint sum.Cauchy productRelated: Its conditional convergence illustrates why Cauchy-product identities need hypotheses.Summability methodCompared with: It shows that alternating terms can yield an ordinary sum without generalized summation.Riemann series theoremBroader topic: It is the standard example whose rearrangements illustrate the theorem.Abel's testBroader topic: Its sign factor has bounded partial sums, while the reciprocal factor decreases to zero.Cauchy's convergence testBroader topic: Its tails can be bounded directly, establishing convergence without absolute convergence.Nth-term testBroader topic: Its vanishing terms pass the test, while the alternating series test establishes convergence.Show all 9