} which logarithms, multiplied by their respective numbers, both true to the last figure. and 14877337702 and the difference of errors 9902681214 Now, since only 6 additional figures are to be obtained, we may omit the three last figures in these errors ; and state thus: as difference of errors 9902681 : difference of sup. 1 :: error 4974656: the correction 502354, which, united to 3,59728, gives us the true value of x = -.3.59728512354. 2. Given 2000, to find an approximate value of x. Ans. x = 4.82782263. 3. Given (6x)* = 96, to find the approximate value of x. Ans, X = 1.8826432. 4. Given wito 123456789; to find the value of x. Ans. 8.6400268. 5. Given 2012 (2x — 2:2), to find the value of x. Ans. X - 1.747933. OF THE BINOMIAL THEOREM. The binomial theorem is a general algebraical expression, or formula, by which any power, or root of a given quantity, consisting of two terms, is expanded into a series; the form of which, as it was first proposed by Newton, being as follows: m т (P + pa)"=p” [1 + 2 + (27") *+ (1.7") )" m т 2n 3n DQ, &c. 4n When p is the first term of the binomial, Q the second * The correct answer to this question has been first given by Doctor Adrain, in his edition of Hutton's Mathematics, who plainly proves that Hutton's answer, which is the same as Bonnycastle's, is incorrect. See Hutton's Mathematics, Vol. I. p. 263, N. Y. Edition. Ed. n term divided by the first, the index of the power, or root, and A, B, C, &c., the terms immediately preceding those in which they are first found, including their signs + or Which theorem may be readily applied to any particular case, by substituting the numbers, or letters, in the given example, for P, Q, m, and n, in either of the above formulæ, and then finding the result according to the rule.* EXAMPLES. 1. It is required to convert (az + x)# into an infinite series. * This celebrated theorem, which is of the most extensive use in algebra and various other branches of analysis, may be otherwise expressed as follows:- a [1+ &c.] )+ m mton m mtnm tan 2t 2m Зп natal at +3 n m 2 n ma m mtn m+2n lata 2a 3], a atal 2. vatal 2n In latal &c. It may here also be observed, that if m be made to represent any whole, or fractional number, whether positive or negative, the first of these expressions may be exhibited in a more simple form, m(m-1) 1-2 m(n-1)(M—2) m-3: (a +x)" =a +-ma x+ 22+ 1.2.3 1.2.3.4 Where the last term is called the general term of the series, because if 1, 2, 3, 4, &c., be substituted successively for n, it will give all the rest. m x2 Х т n 1 2 Х X Х a 2.4a3 323 X 6 a2 2.4.6a5 x х: 3.5804 + Е, 4n 8 2.4.6.807 3.5.24 3.5.7205 X 5n 10 2.4.6.807 a2 2.4.6.8.10a9 &c. &c. &c. 3.5.735 &c. 2.403 2.4.6a5 2.4.6.87 2.4.6.8.10 Where the law of formation of the several terms of the series is sufficiently evident. 1 2. It is required to convert or its equal (a + b)--, (a+b) into an infinite series. m Here P = a, Q= and 2, and n=1: n whence 22 十分 + 2a X Х a 2n m m 2 26 B, 12 a m n 2 - 1 26 b 352 X C, 2 a d4 463 D, 3 a4 25 2-3 403 b 564 Х E, 5 ab &c. 574 Consequently, + &c. (a + b) a a3 a4 a5 aa 3. It is required to convert or its equal aa (az x}}' ( as - x) into an infinite series. Here 1 p=a?, Q and or m ma2 - 1, and n= N 2 1 Х т 1 1 B, a a2 2a3 ec 3х2 C. 4 2a3 a2 2.4a 3.5703 D, 3n 6 aa 2.4.647 1 - C 3.5.724 Х E, 4n 8 2.4.6a7 a2 2.4.6.80 &c. 003 3.5.7 + L ? a 2.4 las 2.4.6 2.4.6.8 i 3 () +, de 十, 4. It is required to convert $79, or its equal (8 + 1)), into n in in 2 = A, m n n an infinite series. 1 1 Here p= 8,Q and or m= 1 and n=3; 3' Whence, 1 рп (8)3n= 8 1 2 1 1 AQ Xi? х B, 23 3.22 m 1 3 1 1 1 X 4 = C, 3.6.24 1 5 Х 3n 9 23 3.6.9.27 5 5.8 E 12 3.6.9.27 23 3.6.9.12.210 m -4n 1-12 5.8 1 5.8.11 EQ= Х 5n 15 3.6.9.12.210 3.6.3.12.15.213 &c. &c. &c. Therefore, /9= 5.8.11 + 3.22 3.6.24 3.6.9.27 3.6.9.12.210 3.6.9.12.15.2139 &c. 5. It is required to convert 72, or its equal ✓ (1+1), into an infinite series. 1 1 3 3.5.7 + &c. 2 2.4 2.4.6 2.4.6.8 2.4.6.8.10 6. It is required to convert 5/7, or its equal (8 – 1)}, into an infinite series. 1 5.8 3.22 3.6.2 3.6.9.27 7. It is required to convert V 240, or its equal (243 — 3)), into an infinite series. 1 4.9.14 Ans. 3 &c. 5.33 5.10.37 5.10.15.311 5.10.15.20.315 2 를 8. It is required to convert (a + x)" into an infinite series. 202 3.23 3.5x4 + &c. 2.4.6a3 2.4.6.8a4 3.6.9.12.210 -, &c. Ans. a *{1 &c. 1 9. It is required to convert (a +b)3 into an infinite series. 3 282 2.563 2.5.864 +, &c. За 3.6a2 3.6.903 3.6.9.12a4 b. Ans. a } 10. It is required to convert (a - b)# into an infinite series. 1+ 1 b: 362 3.763 3.7.1184 Ans. a4 1 &c. 4a 4.8a2 4.8.123 4.8.12.1624 을 11. It is required to convert (a + x) into an infinite series. 13 x2 4203 4.724 4.7.10.25 1+ + За 9a2 923 92.12a4 92.12.1505 Ans. a + 2x ,.&c. . 12. It is required to convert (1 - x)into an infinite series. 2.3x2 2.3.8x3 2.3.8.1324 1 13. It is required to convert or its equal (a + x) (a + x) into an infinite series. 3х2 3.5003 3.5.724 Ans. 2.4.6a3 2.4.6.844 |