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01
THÉ MANNER OK FINDING FLUENTS. 79
■i -. be taken rr a, no number of terms will be fuffieieut to ex-
ax
hibit the value of the correfponding fraction ——, it being
_X ' " "" iM
infinite in that circumftance.
94. Having endeavoured to fhew that the true value of an
infinite feries may be nearly obtained by adding together a few
of the firft terms only, I fliall now proceed to give other examples of the manner of converting fractional and furd quantities into fuch kinds of feriefes, in order to the approximation of the fluents of expreflions affected by them.
EXAMPLE 11.
cz
Let the quantity propofed be the fraclion » -—r
then, by proceeding as in the firft example, you will have
2// 3yz 4y3
cz 4- 2cy 4- yj cz ( 1 - f + "f— ~f
cz 4- 2cy 4- yz
i
Sec.
■—2cy — y*
— 2cy — 4f
2if
2y3
4- 3yz 4- -f Sec.
Where, from a few of the firft terms of the quotient, the
law of continuation is manifeft ; the numerators beinsr in
arithmetical progreflion, and the figns -f and—•alternately.
EXAMPLE III.
1 4- xz — 2i+
95. Let the quantity given be t*
Then the quotient will be 1 4-^4- 3xz 4- 4a-3 4- dx* 4-
9as 4- 14a:6 Sec. where the law of continuation is manifeft,
being fuch that the coefficient of each fucceeding term is
equal to the fum of thofe of the two terms immediately preceding it.
EXAMPLE IV.
96. Let the radical quantity \/az 4- xz be propofed.
Here, according to the common method of extraétioe the
fquare root, tfie procefs will ftand as follows :
| Title | Doctrine and application of fluxions. |
| Alternative Title | The doctrine and application of fluxions containing (besides what is common on the subject) a number of new improvements in the theory, and the solutions of a variety of new and very interesting problems in different branches of the mathematics... To which is prefixed an account of his life. The whole revised and carefully corrected by William Davis. |
| Reference Title | Simpson, Thomas, 1805, The doctrine and application of fluxions. |
| Creator |
Simpson, Thomas, 1710-1761. Davis, William, 1771-1807 . |
| Subject | Calculus. |
| Publisher | London. |
| DateOriginal | 1805 |
| Format | JP2 |
| Extent | 31 cm. |
| Identifier | 138 |
| Call Number | QA303.S45 1805 |
| Language | English |
| Collection | History of Mathematics |
| Rights | http://www.lindahall.org/imagerepro/ |
| Data contributor | Linda Hall Library, LHL Digital Collections. |
| Type | Image |
| Title | Page 79. |
| Format | tiff |
| Identifier | 1138_107 |
| Relation-Is part of | Is part of: The doctrine and application of fluxions containing (besides what is common on the subject) a number of new improvements in the theory, and the solutions of a variety of new and very interesting problems in different branches of the mathematics... To which is prefixed an account of his life. The whole revised and carefully corrected by William Davis. |
| Rights | http://www.lindahall.org/imagerepro/ |
| Type | Image |
| OCR transcript | 01 THÉ MANNER OK FINDING FLUENTS. 79 ■i -. be taken rr a, no number of terms will be fuffieieut to ex- ax hibit the value of the correfponding fraction ——, it being _X ' " "" iM infinite in that circumftance. 94. Having endeavoured to fhew that the true value of an infinite feries may be nearly obtained by adding together a few of the firft terms only, I fliall now proceed to give other examples of the manner of converting fractional and furd quantities into fuch kinds of feriefes, in order to the approximation of the fluents of expreflions affected by them. EXAMPLE 11. cz Let the quantity propofed be the fraclion » -—r then, by proceeding as in the firft example, you will have 2// 3yz 4y3 cz 4- 2cy 4- yj cz ( 1 - f + "f— ~f cz 4- 2cy 4- yz i Sec. ■—2cy — y* — 2cy — 4f 2if 2y3 4- 3yz 4- -f Sec. Where, from a few of the firft terms of the quotient, the law of continuation is manifeft ; the numerators beinsr in arithmetical progreflion, and the figns -f and—•alternately. EXAMPLE III. 1 4- xz — 2i+ 95. Let the quantity given be t* Then the quotient will be 1 4-^4- 3xz 4- 4a-3 4- dx* 4- 9as 4- 14a:6 Sec. where the law of continuation is manifeft, being fuch that the coefficient of each fucceeding term is equal to the fum of thofe of the two terms immediately preceding it. EXAMPLE IV. 96. Let the radical quantity \/az 4- xz be propofed. Here, according to the common method of extraétioe the fquare root, tfie procefs will ftand as follows : |
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