series circuit: see electric circuit.

series, in mathematics, indicated sum of a sequence of terms. A series may be finite or infinite. A finite series contains a definite number of terms whose sum can be found by various methods. An infinite series is a sum of infinitely many terms, e.g., the infinite series 1/2 + 1/4 + 1/8 + 1/16 + … . The dots mean that the remaining terms are formed according to the rule made evident by the first few terms, in this case doubling the denominator of the preceding term to form that of the next term; the nth term of this series is (1/2)ⁿ. Some infinite series converge to a certain value called its limit; i.e., as one adds together progressively more terms, these sums (called the partial sums of the series) form a sequence of values that progressively approach the limit. For example, the series given above converges to the value 1 because the partial sums form the sequence 1/2, 3/4, 7/8, 15/16, … . Many series, however, do not converge, i.e., have no value that their partial sums approach. Such a series is 1/2 + 1/3 + 1/4 + … , for even though the terms become very small, enough of them added together will give a value greater than any number that can be named. A series that does not converge is said to diverge; various tests exist for determining whether or not a given series converges and for determining its limit if it does converge. See also progression.

replacement series: see electromotive series.

paraffin series: see alkane.

methane series: see alkane.

lanthanide series, a series of metallic elements, included in the rare-earth metals, in Group 3 of the periodic table. Members of the series are often called lanthanides, although lanthanum (atomic number 57) is not always considered a member of the series. The series always includes the 14 elements with atomic numbers 58 through 71, which are (in order of increasing atomic number) cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium. All of the members of the series very closely resemble lanthanum and one another in their chemical and physical properties; as the atomic number increases in this series, added electrons enter the 4f electron orbital. Chemically, they are about as reactive as calcium. They all form trivalent compounds; some also form divalent or tetravalent compounds. The lanthanides have similiar sizes so that any given compound of the rare earths is likely to crystallize with the same structure as any other rare earth; however, the lattice parameters become smaller and the crystal denser as compounds proceed across the series.

ethylene series: see alkene.

electromotive series, list of metals whose order indicates the relative tendency to be oxidized, or to give up electrons (see oxidation and reduction); the list also includes the gas hydrogen. The electromotive series begins with the metal most easily oxidized, i.e., the metal with the greatest electron-donating tendency, and ends with the metal least easily oxidized. The tendency to be oxidized is not an absolute quantity; it can only be compared with the tendency of some other substance to be oxidized. In practice, the tendency to be oxidized, called the oxidation potential and expressed in volts, is measured relative to a standard hydrogen electrode, which is arbitrarily assigned an oxidation potential of zero. The oxidation potential measures the tendency of the half reaction M → M⁺ⁿ + ne^- to occur, in which some metal M loses n electrons, e^-, and acquires a positive charge of +n. The more positive the oxidation potential, the more readily oxidation takes place. The electromotive series is thus a list of the metals in the order of their tendency to undergo the half reaction. The series is also called the replacement series, since it indicates which metals replace, or are replaced by, other metals (or hydrogen) in compounds. In general, a metal will replace any other metal lower in the series and will be replaced by any metal higher in the series. The order of some common metals in the electromotive series, starting with the most easily oxidized, is: lithium, potassium, calcium, sodium, magnesium, aluminum, zinc, chromium, iron, cobalt, nickel, lead, hydrogen, copper, mercury, silver, platinum, and gold. A list arranged according to oxidation potential and including not only metals but also all other elements and ions is called the electrochemical series.

actinium series, natural radioactive decay series beginning with uranium-235 (also called actinouranium) and ending with lead-207. See radioactivity.

actinide series, a series of radioactive metallic elements in Group 3 of the periodic table. Members of the series are often called actinides, although actinium (at. no. 89) is not always considered a member of the series. The series always includes the 14 elements with atomic numbers 90 through 103. The other members are (in order of increasing atomic number) thorium, protactinium, uranium, neptunium, plutonium, americium, curium, berkelium, californium, einsteinium, fermium, mendelevium, nobelium, and lawrencium. Thorium and uranium are the only actinides found in the earth's crust in appreciable quantities, although small amounts of neptunium and plutonium have been found in uranium ores. Actinium and protactinium are found in nature as decay products of some thorium and uranium isotopes. All the others have only been synthesized in small quantities (see synthetic elements).

Study of the properties of the actinides is hampered by their radioactive instability. It is known, however, that all members of the series resemble actinium and each other in their chemical properties and that they have a strong chemical resemblance to their homologs in the lanthanide series. The actinides are reactive and assume a number of different valences in their compounds. As the atomic number increases in this series, added electrons enter the 5f electron orbital. Elements in this series with atomic numbers greater than that of uranium (92) are called transuranium elements. Elements with atomic numbers greater than 103 are not members of the actinide series; element 104 (rutherfordium) is the first of the transactinide elements.

acetylene series: see alkyne.

Paschen series: see spectrum.

Lyman series: see spectrum.

Balmer series: see spectrum.

Any of four sets of unstable heavy atomic nuclei that undergo a series of alpha decay and beta decay until a stable nucleus is achieved. The natural series are the thorium series, the uranium series, and the actinium series. These are headed by naturally occurring species of unstable nuclei that have half-lives comparable to the age of the earth. The fourth set, the neptunium series, is headed by neptunium-237, which has a half-life of 2 million years. Its members do not occur naturally but are artificially produced by nuclear reactions and have short half-lives.

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In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. Mathematical analysis is largely taken up with studying the conditions under which a given function will result in a convergent infinite series. Such series (e.g., the Fourier series) are particularly useful in solving differential equations.

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Any of the series of 15 consecutive chemical elements in the periodic table from actinium to lawrencium (atomic numbers 89–103). All are radioactive heavy metals; and only the first four (actinium, thorium, protactinium, and uranium) occur in nature in appreciable quantities. The other 11 (the transuranium elements) are unstable and are produced only artificially. Actinides are transition elements, so their atoms have similar configurations and similar physical and chemical behaviour; the most usual valences are 3 and 4.

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