562 A L G A L G [SERIES. bo less than u n+1 , the only positive term in it. But ?/ B+1 has for its limit, therefore the series is convergent. Ex. 3. 1 - - + - - . . . is convergent, for the sum of I- O the series after the nth term is less than - , which has n+1 as its limit. Prop. 3. If the terms of the series are all positive, and the limit of the nth term is ; then if the limit of the quotient of the (n+l)th term by the nth be less than 1, the series is convergent ; but if the limit be greater than 1, the sum is divergent. 1. Let & be the greatest value of , after a certain u n value of n, and k < 1 ; then, which has for its limit. Hence the series is convergent (Prop. 1). 2. Let L the least value of -^ after a certain finite M value of n, be greater than 1 ; then u n+l = or >ku n n+2 = or >7t 2 ?/ n &c. <fcc. u n+1 + u n+z + &c.= or >ku n (l+k + Jc z + &c.), which is infinite. Hence the series is divergent. Prop. 4. If - - be less than 1 ; then the two series are both convergent, or both divergent together. Series (2) + ?^ = 2( 1 + ?<, + 2 4 + <iu s + . . .), which is equal to or less than the following, term by term, viz. : 2 {j + u. 2 + (?/ 3 + ?* 4 ) + (u b + u 6 + n. + u s ) + . . . } , i.e., twice series (1). Hence if the one series be convergent, the other will be also convergent; and if series (2) be divergent, series (1) is also divergent. Again, series (2) is equal to or greater than the follow ing, term by term, which is series (1). Hence if series (1) be divergent, series (2) is also divergent. Ex. 1. The series + + r- + . . . is convergent if Zi r O r ?>!, but divergent if r= or < 1. The two series (1) and (2) now become the latter of which is the geometric series 1 1 1+ 2 r-l + 4^1 + which is convergent or divergent according as r > 1 or the contrary. Hence the same is true of the given series. Ex. 2. The binomial series 1 + nx + etc., is convergent when x<, divergent when x> . Ex. 3. To find when the binomial series l-n + n n ~ &c., is convergent. Let n< 1; the (?+!) term may be written n rnl rn2 r - 2 l__!i r-l) V r-2 1 2 r-l .J . (Art. 122, Ex. 7) rVr-l r-2 whence (Prop. 4, Ex. 1) the series is convergent. Similarly in other cases. (P. K.) ALGECIRAS, or ALGEZIRAS, a seaport of Gpain, in the province of Cadiz, 6 miles W. of Gibraltar, on the opposite side of the bay. The town is picturesquely situated, and its name, which signifies in Arabic the island, is derived from a small islet which forms one side of the harbour. It is supplied with water by means of a beautiful aqueduct. It has a dilapidated fortress, and also a military hospital. Though the harbour is bad, and the commerce of the town has considerably declined, there is still a good coasting trade; the exports and imports averaging about 60,000 annually. Charcoal and tanned leather are the chief articles >f export. Algeciras was the Portus Albiis of the Romans, and the first place in Spain taken by the Moors. It remained in their possession from 713 till 1344, when it was taken by Alphonso XI. of Castile after a celebrated siege of twenty months, which attracted crusaders from all parts of Europe, among whom was the English earl of Derby, grandson of Edward III. It is said that during this siege gunpowder was first used by the Moors in the wars of Europe. The Moorish city was destroyed by Alphonso, and the modern town was not erected till 1760. During the siege of Gibraltar in 1780-82. Algeciras was the station of the Spanish fleet and floating batteries. Near Algeciras, on 6th July 1801, the English admiral Saumarez attacked a Franco-Spanish fleet, and sustained a reverse; but on the 12th he again attacked the enemy, whose fleet was double his own strength, and inflicted on them a complete defeat. Population, 14,000. ALGER OF LIEGE, known also as ALGER or CLUGNY and ALGERUS MAGISTER, a learned French priest who lived in the first half of the 12th century. He was first a deacon of the church of St Bartholomew at Lie ge, his native town, was afterwards translated to the cathedral church of St Lambert, and finally retired to the monastery of Clugny, where he died not later than 1145, though the precise date is uncertain. His History of the Church of Liege, and many of his other works are lost. The most im portant of his still extant works are: 1. De Misericordia et Justitia, a collection of extracts from Fathers, with reflections, which is to be found in the Anccdota of Martene, vol. v. 2. De Sacramento Corporis et Sanguinis Domini; a treatise, in three books, against the Bercngarian heresy, highly commended by Peter of Clugny and Erasmus. 3. De Libero Arbitrio; given in Pez s Anecdota, vol. iv. 4. De Sacrijicio Missce; given in the Collectio Scriptor. Vet. of Angelo Mai, vol. ix. ALGERIA, or ALGIERS (French, L Algerie), the largest and most important of the French colonial possessions, is a country of Northern Africa, bounded on the N. by the
Mediterranean, W. by the state of Marocco, S. by the
