Search results
Results from the WOW.Com Content Network
A classic example is the alternating harmonic series given by + + = = +, which converges to (), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem .
If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series . Many standard tests for divergence and convergence, most notably including the ratio test and the root test , demonstrate absolute convergence.
If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
otherwise the test is inconclusive (the series may diverge, converge absolutely or converge conditionally). There are some series for which C = 1 and the series converges, e.g. ∑ 1 / n 2 {\displaystyle \textstyle \sum 1/{n^{2}}} , and there are others for which C = 1 and the series diverges, e.g. ∑ 1 / n {\displaystyle \textstyle \sum 1/n} .
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence. [4] Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The general principle is that addition of infinite sums is ...
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.