SERIES. 



Tims, of the former kind, we have 



b b' 



I 



■ + ■ , 



a- a' 



-1—- {a + i)- = — , + 



a -i b a a 



b 



J + &c. 



I I " b' b' ^ 



a - i ^ '■ a a- a' a* 



and a variety of others. And of the latter. 



a 



+ 



a 



-f Sec. 



+ 



I I I » „ 



S:c. &c. &c. 



Many of which are Hill irreducible to any equivalent finite 

 funftion. 



Series alfo receive feveral different denominations ac- 

 cordiiii^ to certain circumftances attending their formation, 

 the law which they follow, the form of the fnnftion to 

 which they are reducible, S:c. Si.c. as arithmetical, geome- 

 trical, converging, diverging, reciprocal, &c. fcries. 



Series, Converging, are thole in which the terms decreafe, 

 or become fucceflively lels and lef< ; as 



are circular feries ; the former being equal to one-eighth of 

 the circumference of a circle whofe radius is i ; and the 

 latter equal to one-fixth of the fquare of the femi-circum- 

 ference to the fame radius. 



Series, Logarithmic, are thofe whicli exprefs, or whofe 

 fums depend upon the logarithms of numbers ; as 

 (a-i)-l(a-iy + ^{a-iy-i(a- l)> + Scc. 

 which is equal to the hyperbolic logarithm of a. 



SERrES, Arit'iimelical, are thofe whofe fucceflive terms dif. 

 fer from each other by a certain and determinate quantity ; as 



a + (<j -I- </) -I- (a 4- 2J) -t- (a + 3J) -I- &c. 

 a -h (a — </) + (a — 2(/) + (a — 3</) -(- &c. 



Series, Giometrical, are thofe whofe fucceflive terms are 

 fome multiple or fubmultiple of thofe immediately preceding 

 them ; as 



a + ra + r' a + r^a + r' a + &c. 



a + — + 



r 



- - -I- — -t- &c. 

 r ' r' 



I I 



1 + — + "i 

 5 S 



Series, Diverging, are thofe in which the terms con- 

 tinually increafe ; as 



Series, FraSioiuil, are thofe whofe terms arc all frac> 

 tional ; as 



a a a 



b{b ^ + {b + c)(i+2c) + (b+2c){b+y) "^ '"• 



Series, Trigonometrical, are thofe which relate to trigo- 

 nometrical lines or quantities; as 



fin. ' a ^ fin.* a 



fin 



i fin.* a i . c fin.'' a 



a + — -i- -■' — -I- ^ ^^ -I- &c. 



2. 3r' 2 . 4 . 5r' 2.4.6. yr 



z + 2' 



+ 2* 



&c. 



tan. a 



tan. ' a 



"7^ 



+ &c. 



Series, Neutral, are thofe in which all the terms are ^hich are each expreffions for the length of a circular arc, 



equal to each other ; as the former in terms of the fine, and the latter in terms of 



the tangent. 

 '~' + '~'+ I — 1 + &c. Series, Exponential, are thofe which arife from the 



This arifes from the divifion of 1 by i + i, and is there- <^M«"!>:-'n of. or whofe fum depends upon exponential 



fore equal to i. qujnt.t.es ; as 

 Series, Indeterminate, is fometimes ufed to denote a 



feries, whofe terms proceed according to the powers of fome 

 indeterminate letter or quantity ; as 



I + 



X 



I 



+ — + - 

 2 2 



-3-^^ 



3-4 



-H &c. 



* + 



+ 



+ 



+ - - .V 



5 



which is equal to e' , c being the number whofe hyperbolic 

 logarithm is i . 

 + ^c. Series, Recurring, arc thofe in which each term has a 



conllant relation to a certain number of the preceding terms. 

 Other writers, however, mean by this denomination thofe See Recur Ki NG 5f/-/«. 



feries whofe fums are indeterminable in any finite form. Series, Law of a, is ufed to denote that relation which 



Series are again either afcending or defcer.ding. fuhfills between the fucceflive terms of a feries and by 



Series, Afcending, are thofe in which the powers of the which their general term may be denoted : thus the feries 

 indeterminate quantity continually increafe ; as 



1 + flj: -I- b x"- -I- cx^ 4- dx* + &c. 



I + - .V 



3 



'5 



16 



+ ' -v 



35 



'28 • , 



+ 3C' -I- &C. 



3'5 



Series, Defcemling, are tliofe in wliich thefe powers de- "^^y be put under the form 



creafe in the numerator, or increafe in the denominator ; as 



1 + as: 



I + 



-I- bx- 



X 



1 + 



. 2.4 



+ -- .r- 



!•• + — ^ X* + So. 



+ 



+ ex 



H 



+ dx 

 d 



* + &c. or 

 + &c. 



2.4.6 



3 3-5 3^5-7 3-5-7-9 



where the law by which it may be indefinitely continued is 

 manifeit ; and from which we draw the general term, vii. 



2.4.6.. . 2 (» - i) 



Series, Circular, are thofe whofe fums depend upon the 3.5.7... (2n— i)" 

 qvadrature of the circle • thus, „ , , ■ r r, ■, 



oeRIES, Interpolation of See INTERPOLATIOK. 



I I I 1 Series, Revcr/ion of. See Reversion. 



3^ 5 ~ 7^ "*^ ^^ ^ Series, Summation of, is the finding the fum of a feriet, 



whether the number of its terms be finite or intii.ite ; the 

 I^JL^_I_^.J |.J jjp various methods of performing which is treated of in the 



3' 



fubfequent part of this article. 



Mtthod 



