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# Serampore

## series, term, terms, constant, quantity, called and multipliers

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'SERAMPORE, a town of Bengal belonging to Denmark. It is agreeably situated on the west bank of the Bhagarutti or Hoogly river. The territory which belongs to it is about a mile long and half a mile broad, stretching along the banks of the river. The houses are seldom above two stories high, and are built of brick, and plastered with mortar. They have balconies, Venetian windows, and flat roofs. A handsome church is the principal public building. The town is not fortified; but there is near the flagstaff a battery of 12 pieces of cannon. A very trifling trade is carried on between this place and China and Europe. Being a sanctuary for creditors, which are British subjects, it is principally supported by them and by the missionaries. East long. 88° 26'. North lat. 22° 45'.

SrAtiEs, in .lna/gsis, is a number of quantities ar ranged in a certain order or succession, and so related that each succeeding quantity may be known from those which precede it.

The quantities which compose a series are called its terms, and that relation which is observable among the terms by which they may he successively deter mined, is called the law of the .series.

Series are denominated, according to the nature cf their terms, numerical or aigebraitai. Thus I, 2, 3, 4, &c.

1 1 1 2 3 4 are numerical series, and x, sl, &c.

b CAC• a' are algebraical series.

The laws of these series arc severally obvious upon inspection, and there is no difficulty in continuing them to any number of terms.

Series are variously denominated in reference to their laws.

An arithmetical series is one in which each term is found by adding to or subtracting from the preceding term the same quantity. Such are the following: 1, 2, 3, 4, Sze.

3, 6, 9, 12, Sze.

a, aid, o +2 d, a+3 d, Scc, a,a—d, a-2 d, a-3 d, &c.

A geometrical series is one in which each term is found by multiplying the preceding term by the same quantity. Such are the following: 2, 4, 8, 16, fee.

a, (yr, ((xi, Sce.

2 being the multiplier in the former, and x in the latter.

An harmonical series is one in which the reciprocal of each term is found by adding to or subtracting from the reciprocal of the preceding term the same quantity. Such are

1 1 , 1,..cc. 2 0 4 , a , c et a+d a + 2d ad-3d the quantity added being I in the former, and the latter.

These three series are commonly called progres sions. For a detailed account of their properties, see AnGr.nitn.

A recurring series is a general class, of which a geometrical series is a particular example. As each term of a geometrical series is produced by multi plying the preceding term by a constant quantity', so in a recurring series each term is found by multiply ing a certain number of the terms which immediately precede it by as many constant quantities. Thus let A and Ii be any two successive terms of a recurring series, and let a and b be the constant multipliers, the next term will be a A -4- b B. Let this be called C, that is, let a A + b 13 =-- C; then the succeeding term is a 13 b C. Again, let a 13 C=, ll, and the following term is a C b D, and so on.

In like manner, if the series be produced by three constant multipliers, each term may be found from the three terms which immediately precede it. Let A, B, and C, be the three consecutive terms, a, b, c, the three constant multipliers, and let I), E, F. &c. be the terms which immediately succeed C. Then we have and the process would be similar if the series were generated by four or more constant multipliers.

In a rect.rring series the system of constant multi pliers is called the scale of relation, and the series is said to he a recurring series of the first, second, third, or lath order, according as the number of constant multipliers in the scale of relation is 1, 2, 3, or ra.

It is evident that a recurring series of the first order is a geometrical series.

When powers of the same letter or species enter all the terms of a series as factors, the series is called an ascending or descending series with respect to this quantity, according as the exponents of the power increase or decrease. Thus the following is an as cending series with respect to x, and a descending one with respect to a.

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