391. Another method of determining the convergency or divergency of series is to find the limit of the sum of the series after the first n terms; which also determines the limits of the error arising from taking any number of terms instead of the whole series. Thus, +...... the sum of the series But this quantity decreases as n increases; and, by increasing n without limit, it may be made less than any assignable quantity. Therefore the series is convergent. And if n terms be taken for the whole series, the error is less than and, since the quantities within the small brackets are all positive, this sum both which quantities are diminished without limit as n is increased. Hence the series is convergent; and if n terms be taken for the whole series, the error is less than the n+1th term. 1 1 1 Ex. 3. In the series 1+ + + +.. n terms 2 3 4 be 392. In the series a‚x+a,x2+a‚x3+...... in inf. such a value may given to x, that the value of the whole series shall be less than any proposed quantity p. Let be the greatest of the coefficients a1, Agı Az, whole series is less than then the hence that which is required is done, if x be such a value that COR. Hence also in the series a+a1x+ax2+...... in inf. such a value may always be given to x, that the first term is greater than the sum of all the other terms. This value of x will be any quantity less 393. RECURRING SERIES. If each succeeding term of a decreasing Infinite Series bear an invariable relation to a certain number of the preceding terms, the series is called a Recurring Series, and its sum may be found. Let a + bx + cx2+ &c. be the proposed series; call its terms A, B, C, D, &c. and let C=ƒxB+gx2A, D=fxC+gx2B, &c. where f+g is called the scale of relation; then, by the supposition, A = A, B=B, C=fxB+gx2A, D=fxC+gx2B, E=fxD+gx°C, &c. = &c. and, if the whole sum A+B+C+D+&c. in inf. = S, we have S=A+B+fx.(S − A) + gx2. S, or S-fxS-gx2S = A + B −ƒxA; In the same manner, if the scale of relation be f+g+ &c. to n terms, the sum of the series is A+B+C...ton terms-fr{ A+ B...ton-1 terms} -ga2 { A+...ton-2 terms} -&c. 1-fx-gx3-hx3...to (n+1) terms Ex. 1. To find the sum of the infinite series 1+3x+9x2+ &c. when a is less than . 1 Here f= 3, and the sum = 1-3x Ex. 2. To find the sum of the infinite series 1+2x+3x2 +4x+&c. when a is less than 1. Here f=2, g=-1, and 1+2x-2x the sum = 1 = If = or >1, the series is infinite; yet we know that it arises from the division of 1 by (1-x), and the sum of n terms may be determined. The series after the first n terms becomes (n + 1)x" + (n + 2)x"+1 + (n+3)x”+2+ &c. in which the scale of relation, as before, is 2-1; and therefore the series arises from the fraction (n + 1)x" + (n + 2)x2+1 − 2(n+1)x2 +1 or 1-2x+x2 (n+1)r" - np"+1 .. 1+2x+3x2+ &c. to n terms = 1−(n+1)x+ng Cor. If the sign of x be changed, 1-2x+3x3- &c. to n terms 1=(n+1) "=n+1 where the upper or lower sign is to be used, according as n is an even or odd number. Ex. 3. To find the sum of n terms of the series 1+ 3x + 5x2+7x03 + &c. Suppose f+g to be the scale of relation; then 3f+g=5, and 5f+ 3g=7; hence f=2, and g= -1; and, by trial, it appears that the scale of relation is properly determined; After n terms, the series becomes (2n+1)x+(2n+3)x+1+ (2n+5)x+2+&c. hence 1+3+ 5x2+ 7x3+ &c. to n terms, 1+x−(2n+1)x+(2n-1)x+1 (1-x)2 ; Ex. 4. To find the sum of 1+2x+3x2+5x3+8x1+&c. in inf. when the series converges. In this case the scale of relation is 1+1, and consequently the sum is Ex. 5. To find the sum of n terms of the series The scale of relation is 2-1; therefore the sum in inf. is (n−1)x+(n−2) x2 — 2 (n − 1)x2 (n − 1 ) (1-x)2 or − n (1-x)2 After n terms, the series becomes - "+1-2x+2- &c. the sum of which is found in the same manner to be (1-x)2 ; |