Ans. 2x —2x+7x1—13**+34x3, &c. PROBLEM 48 -, or; two terms are deficient by; three terms exceed it by; four terms are deficient by ; five terms will exceed the truth by, &c. So that each succeeding term of the series brings the quotient continually nearer and nearer to the truth by one half of its last preceding difference; and consequently the series will approximate to the truth nearer than any assigned number or quantity whatever; and it will converge so much the swifter, as the divisor is greater than the dividend. But the second series perpetually diverges from the truth ; for the first term of the quotient exceeds the truth by I-, or ; two terms thereof are deficient by ; three terms exceed it by ; four terms are deficient by ; five terms exceed the truth by 37, &c. which shew the absurdity of this series. For the same reason, x must be less than unity in the second example; if x were there equal to unity, then the quotient would be alternately 1, and nothing, instead of ; and it is evident, that x is less than unity in the first example, otherwise the quotient would not have been affirmative; for if x be greater than unity, then 1-x, the divisor, is negative, and unlike signs in division give negative quo tients. From the whole of which it appears, that the greatest term of the divisor must always stand first. PROBLEM II. To reduce a compound surd to an infinite series. RULE. Extract the root as in Arithmetic, and the operation, continued as far as may be thought necessary, will give the series required. EXAMPLES. 1. Required the square root of a'x' in an infinite -series. * This rule is chiefly of use in extracting the square root ; the operation being too tedious, when it is applied to the higher powers, Here the square root of the first term, a2, is a, the first term of the root, which, being squared and taken from the given. surd a2+x2, leaves 2; this remainder, divided by 2a, twice the 2. Required the square root of a-x in an infinite series. x Ans. a &c. za 8a3 16as 3. Convert 1+1 into an infinite series. Ans. 1+-+-, &c. 4. Let xx be converted into an infinite series. 3 5. Let 1-3 be converted into an infinite series. the two the root, which must be added to the double of at 2a And by first terms of the root, for the next compound divisor. proceeding thus, the series may be continued as far as is desired. NOTE. In order to have a true series, the greatest term of the proposed surd must be always placed first. SIMPLE EQUATIONS. AN EQUATION is when two equal quantities, differently expressed, are compared together by means of the sign placed between them. Thus, 12—5—7 is an equation, expressing the equality of the quantities 12-5 and 7. A simple equation is that, which contains only one un known quantity, in its simple form, or not raised to any power. Thus, x-abc is a simple equation, containing only the unknown quantity x. Reduction of equations is the method of finding the value of the unknown quantity. It consists in ordering the equation so, that the unknown quantity may stand alone ‚on one side of the equation without a coefficient, and all the rest, or the known quantities, on the other side. RULE 1.* Any quantity may be transposed from one side of the equation to the other, by changing its sign. Thus, if x+3=7, then will x=7—3=4- RULE *These are founded on the general principle of performing equal operations on equal quantities, when it is evident, that the results must still be equal; whether by equal additions, or sub tractions, or multiplications, or divisions, or roots, or powers. RULE 2. If the unknown term be multiplied by any quantity, that quantity may be taken away by dividing all the other terms of the equation by it. Thus, if axaba, then will x1. And if 2x+4=16, then will x+2=8, and x—8—2 6. In like manner, if ax+2ba3c, then will x+26= If the unknown term be divided by any quantity, that quantity may be taken away by multiplying all the other terms of the equation by it. Thus, if=5+3, then will *=10+6—19. And, if bed, then will xabac-ad. 2x In like manner, if 26+4, then will 2x-6 3. 18+12, and 2x18+12+6=36, or x—3—18. RULE 4. The unknown quantity in any equation may be made free from surds by transposing the rest of the terms according to the rule, and then involving each side to such . power, as is denoted by the index of the said surd. Thus, if √x-2=6, then will *8-64. x=6+2=8, and And, if ✔ 4x+16=12, then will 4*+16=144, and 128 4x144-16128, or x=-=32. 4 In 1 |