(37.) The two following series will be found very useful in rendering the general term of a series integrable. a Pm (Us+am) n+1 hn-m+1 m-1 =S. m n+1 m n-s+1,n P, (u2+ r− 1) · S ̧ (−1)m+* . ▲m−1¤a−1. C. (ar), (38.) Recurring series. The general form of a recurring by the integration of which the general term may be obtained, and thence the sum of the series. By the following method the sum of a series may be obtained from its equation, without knowing the general term: for a write +1, and substitute Uz + n − m + Az Uz + n − m then (1) becomes for u2+n-m+1? 0 = a ▲zUx+n+ (a + a1) ▲ ̧U2+n−1 + &c. +(a + à1 + &c. + an−1) ▲ xux + 1 + (a + a1 + &c. +an)uz+ · Eu2+1=const. au2+n+ (a + a1) U2+n−1 + &c. + (a + a1 + &c. + an − 1) Uz + 1 a + a1+ &c. + an (Tr. L. App. 390—5.)` (39.) Application of the integral calculus to series. The sum either of an infinite or a limited number of terms of many series may be represented by a definite integral: the following methods are applicable to numerous classes of series, consisting of ascending powers of some quantity, the coefficients of which are composed of arithmetical factorials. Let the series be s=at" + (a+b)tm+n+(a+2b) tm + 2n + &c. by differentiating which, the value of s may be obtained. · If s=(a+b)(c+e)TM+ (2a+b)(2c+e)tm+n+ &c. +(ax+b)(cx + e) ¿m + (x − 1) x from which s may be obtained, after two differentiations. The same method may be applied to the series of which the ath term is (a ̧x+b1)(a2x+b2).......(ax+b ̧) TMTM+ (x − 1)n ̧ multiplying this by ct, and differentiating, and then multiplying Let s=(a+b)t + (a+b)(2a + b) t2 + &c. +(a+b)(2a+b).....(ax+b)ťo, 1 -1 then multiplying by-ta , we obtain the equation α at d,s+{(a+b)t−1}s=(a+b)...(x+1.a+b)ť+1— If the th term is (a+b)t. (a+b)(2a+b).....(ax+b), then mul (c + e)(2c + e)...(cx +e) and integrating, and then multiplying the quantity thus obtained by ct, and differentiating, the result is (Tr. L. App. 412—5; L. C. D. 1140—8.) THEORY OF GENERATING FUNCTIONS. (40.) Let the indefinite series be represented by (t): this is the generating function of u The generating function of u2+ is t−".p(t), from the development of t-"p(t), when put under the form n {1+ ( − 1)}"p(4), and of (−1)"p(t), the values of u2+n, and Aru, in Art. 2 may be obtained. (41.) Let vu, be used to represent the series au+a12+1+а2 U ̧ + 2 + &c. + αnuz+n the generating function of which is |