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The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. [1] [2] [3] For a generalization, see Dirichlet's test. [4] [5] [6]
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
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]
A scanning transmission electron microscope (STEM) is a type of transmission electron microscope (TEM). Pronunciation is [stɛm] or [ɛsti:i:ɛm]. Pronunciation is [stɛm] or [ɛsti:i:ɛm]. As with a conventional transmission electron microscope (CTEM), images are formed by electrons passing through a sufficiently thin specimen.
Diagram illustrating the light path through a dark-field microscope. The steps are illustrated in the figure where an inverted microscope is used. Light enters the microscope for illumination of the sample. A specially sized disc, the patch stop (see figure), blocks some light from the light source, leaving an outer ring of illumination. A wide ...
Convergent beam electron diffraction (CBED) is an electron diffraction technique where a convergent or divergent beam (conical electron beam) of electrons is used to study materials. CBED scheme, adapted from W. Kossel and G. Möllenstedt, Elektroneninterferenzen im konvergenten Bündel, Annalen der Physik 36, 113 (1939).
The light path of a bright-field microscope is extremely simple; no additional components are required beyond the normal light-microscope setup. The light path begins at the illuminator or the light source on the base of the microscope. Often a halogen lamp is used. The light travels through the objective lens into the ocular lens, through ...
A condenser between the stage and mirror of a vintage microscope. Condensers are located above the light source and under the sample in an upright microscope, and above the stage and below the light source in an inverted microscope. They act to gather light from the microscope's light source and concentrate it into a cone of light that ...