2. Find the sum of 20 terms of the series 1, 4, 10, 20, 35, etc. 3. Find the sum of n terms of the series 1, 2, 3, 4, 5, 6, etc. 372. Interpolation is the process by which, when we have given a certain number of terms of a series, we compute intermediate terms which conform to the law of the series. Interpolation may, in most cases, be effected by the use of the formula of Art. 370. If in this formula we substitute n+1 for n, we shall have Tn+1=a+nD'+ n(n − 1)D" + " 2 n(n−1)(n−2) D'''+, etc., 2.3 which expresses the value of that term of the series which has n terms before it. When n is a fraction less than unity, Tn+1 stands for a term between the first and second of the given terms. When n is greater than 1 and less than 2, the intermediate term will lie between the second and third of the given terms, and so on. In general, the preceding formula will give the value of such intermediate terms. EXAMPLES. 1. Given the cube root of 60 equal to 3.914868, to find the cube root of 60.25. Here D'=+.021629, D".000235, D'"+.000007, etc. a=3.914868, and n=.25. Substituting the value of n in the formula, we have Tn+1=a+1D'—331⁄2D′′+11⁄2¿D′′ etc. 32 The value of the 1st term is 28 +3.914868, + .000000. Ans. 3.925712. Ans. 3.931112. Ans. 3.927874. Ans. 3.922031. Hence the cube root of 60.25 is 2. Find the cube root of 60.5. 3. Find the cube root of 60.75. 4. Find the cube root of 60.6. 5. Find the cube root of 60.33. 6. Given the square root of 30 equal to 5.477226, Development of Algebraic Expressions into Series. 373. An irreducible fraction may be converted into an infinite series by dividing the numerator by the denominator, according to the usual method of division. 1 Hence 1-x 1+x+x2+x3+x2+x+, etc., to infinity. Suppose x, we shall then have Ans. 1+2x+3x2+5x3+8x2+13x5+, etc. 374. An algebraic expression which is not a perfect square may be developed into an infinite series by extracting its square root according to the method of Art. 198. Ex. 1. Develop the square root of 1+x into an infinite series. Hence the square root of 1+x is equal to Ex. 2. Develop the square root of a2+x into an infinite series. 2a 8a3 16a5—128a7+, etc. 4 Ex. 3. Develop the square root of a1 +x into an infinite series. Ex. 4. Develop the square root of a* -x into an infinite series. Ex. 5. Develop the square root of a2+2 into an infinite. series. Method of Undetermined Coefficients. 375. One of the most useful methods of developing algebraic expressions into series is the method of undetermined coefficients. It consists in assuming the required development in the form of a series with unknown coefficients, and afterward finding the value of these coefficients. This method is founded upon the properties of identical equations. 376. An identical equation is one in which the two members are identical, or may be reduced to identity by performing the operations indicated in them. As 377. It follows from the definition that an identical equation is satisfied by each and every value which may be assigned to a lettér which it contains, provided that value is the same in both members of the equation. Every identical equation containing but one unknown quantity can be reduced to the form of A+Bж+Сx2+D3+, etc. A'+B'x+C'x2+D'x3+, etc. |