is the limit. When v is very great, approaches to unity, 1+0 1+v but never will equal it. If v be a fraction, the convergency will be greater, the greater the denominator is. This series is a transformation of p-qv+rv-sv3+&c. and is particularly useful in finding a rapid approximation to such slow con verging series as 1 1 1 1 2+3-4 +&c. or 1 1 1 1 +&c., where, in both cases, v would be equal to unity, and the first term of the series, z=1 in the first, and 2 in the second. being given, to find the limits of convergency. (2n-m)v -&c.) ; therefore the limit Hence the convergency will be rapid when v is small compared with a: when x=3, the convergency in the third term is v(n-m) 2na The series here proposed is the expansion of (a+v). Therefore to find the arithmetical expansion of (a+v), a ought to be a large square number, and v small. Thus, to find the square root of 5; then m=1, n=2, make a =4,v=1, then a=4=2; the above series then be comes, 3 5 2(1+124-42+BC+&c.) 12500 97 1 2)12500 1.12599 8)6250 781 3 3)2343 8) 781 97 4)485 8)121 15 5)105 1.11803 2 2.23606, which is the square root of 5 true to the last figure. And thus any power or root may be extracted. The ultimate convergency depending on the numbers v and a, which is here, and not upon the indices; the series will, however, converge slower, the higher the root is to be extracted. 8) 21 2 Problem 3. A series being given to find the value of a, the first factor, so that it may obtain a given convergency at the ath term. Find the convergency of the series, and for z substitute its value in terms of a, and the first term a; then make the formula, thus altered, equal to the given convergency, and the value of a will be found by this equation. me of a, so that the series may obtain a given convergency at the s term of the series. their term is x(x-1)+a; which, being substituted in the general 2-2 a+z(x-2) convergency z + x(m-1)' gives a+x(m+x-2); then the resolu a+x(x-2) 1 tion of the equation a+x(m+x-2)=gives a= + x(m+x-2)-b(x-2) b-1 Here x=2, m=5, x=4, 6=3; therefore : 2(5+4-2)-2.3(4-2) 3-1 1 3.5.7.9.11 =1; and therefore the series 1 1.3.5.7.9 + &c. will obtain a convergency of f in the fourth term. The series 4 (+)+B(+2) A R -&c. being given, to R find the value of a, so as to make the convergency term. in the th (1) The convergency of this series is expressed by [x(x-1)+a] R Two series which do not converge at the same rate being given, to find the number of the term, if possible, in which their convergencies are equal. Find the convergency of each series; then make the two expres. sions equal; and the value of being found, that of x will be ob tained from the equation x=x(x-1+a. Or, find the convergency of each series in terms of x and a, the first factor, and make the two expressions equal; then the value of x in the equation will give the number of the term. (a+8)R (a+12) R + &c. being given, to find & the number of the term when their convergencies are equal. The convergency of the first of these series is and that of the second is 4x-8+a 9[4(x-1)+a] 4(x-1) R[4(x-1)+a]; therefore, putting these 4(x-1) 4x-8+a expressions equal to one another, thus, R[4(x-1)+a] 9[4(a-1)+a] or 4(x-1)_4x-8+a R 9 aR+36-8R for the number of , gives x the term, as required. Let a=33; then will x=59. THE ELEMENTS OF EUCLID. BOOK I. 1. A POINT is that which hath no parts, or which hath no magni tude. 2. A line is length without breadth. 3. the extremities of a line are points. 4. A straight line is that which lies evenly between its extreme points. 5. A superficies is that which hath only length and breadth. 6. The extremities of a superficies are lines. 7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 8. "A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direc tion." 9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. N. B. When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line: Thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, BD, is named the angle ABD, or DBA; and that which is contained by DB. CB, is called the angle DBC, о. |