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Additional resources for Table of Integrals, Series, and Products, Seventh Edition
Example text
K=1 ∞ 6. ∗ k=1 ∞ EU 1 π2 = 2 (2k − 1) 8 EU (−1)k =G (2k + 1)2 FI II 482 (−1)k+1 π3 = (2k − 1)3 32 EU 1 π4 = 4 (2k − 1) 96 EU (−1)k+1 5π 5 = (2k − 1)5 1536 EU (−1)k+1 7. 236 ∞ 1. k=1 ∞ 2. k=1 ∞ 3. k=1 ∞ 4. 1 = 2 ln 2 − 1 k (4k 2 − 1) BR 51a 3 1 = (ln 3 − 1) 2 k (9k − 1) 2 BR 51a 3 1 = −3 + ln 3 + 2 ln 2 k (36k 2 − 1) 2 k = 2 (4k 2 − 1) k=1 ∞ 5. 3) BR 52 3 − 2 ln 2 2 BR 52 10 Numerical Series and Infinite Products ∞ 12k 2 − 1 6. 237 ∞ 1. k=1 ∞ 2. k=1 ∞ 3. 2), BR 52 1 π 1 = − (4k − 1)(4k + 1) 2 8 3 1 = (k − 1)(k + 1) 4 ∞ 4.
These differences in notation must also be taken into account when using the following relationships that exist between the Bernoulli and Euler polynomials: B n (x) = 1 2n n k=0 n Bn−k E k (2x) k E n−1 (x) = 2n n E n−1 (x) = x 2 B n (x) − 2n B n n 2 Bn x+1 2 − Bn n = 0, 1, . . x 2 or and n = 1, 2, . . n−2 n n −1 2n−k − 1 Bn−k B n (x) n = 2, 3, . . k 2 k=0 There are also alternative definitions of the Euler polynomial of order n, and it should be noted that some authors, using a modification of the third expression above, call 2 x B n (x) − 2n B n n+1 2 the Euler polynomial of order n.
In several cases, the editors have used Russian editions of works published in other languages. Under such circumstances, because the pagination and numbering of equations may be altered, we have referred the reader only to the original sources and dispensed with page and equation numbers. 111 Arithmetic progression. 112 Geometric progression. 113 [l = a + (n − 1)r is the last term] [q = 1] Arithmetic-geometric progression. 121 k=1 n 1. k= k=1 n(n + 1) 2 n k2 = 2. k=1 n k3 = 3. k=1 CE 333 n(n + 1)(2n + 1) 6 n(n + 1) 2 CE 333 2 CE 333 1 2 Finite Sums n 4.