By I S Gradshteĭn; I M Ryzhik; Alan Jeffrey; Daniel Zwillinger
ISBN-10: 0123736374
ISBN-13: 9780123736376
ISBN-10: 0123738628
ISBN-13: 9780123738622
'The desk of Integrals, sequence and items' is a big reference resource for integrals within the English language. it truly is crucial for mathematicians, scientists, and engineers, who depend on it whilst determining and therefore fixing tremendous advanced problems.
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Additional info for Table of integrals, series and products
Sample text
K=1 n JO (32a) (2k − 1)3 = n2 (2n2 − 1) 3. 124 q k n2 − k 2 = 1. 10 k=1 1 q(q + 1) 2n2 − q 2 − q 4 [q = 1, 2, . ] 1 n(n + 1) 12n3 + 63n2 + 107n + 58 60 n k! · k = (n + 1)! 126 k=0 (n + k)! (n − k)! 13111 k=1 1 1 = C + ln n + − k 2n ∞ k=2 Ak , n(n + 1) . . 133 k=2 23 − 1 B4 1 B2 1 = (C + ln n) + ln 2 + 2 + + ... 14 Sums of products of reciprocals of natural numbers n 1. k=1 n 2. k=1 n 1 = [p + (k − 1)q](p + kq) p(p + nq) GI III (64)a n(2p + nq + q) 1 = [p + (k − 1)q](p + kq)[p + (k + 1)q] 2p(p + q)(p + nq)[p + (n + 1)q] GI III (65)a n 3.
1 = ln 2 k (−1)k+1 1 =1−2 2k − 1 ∞ k=1 3. ∗ ∞ k=1 rq a + 1 − q (1 − q)2 (−1)k+1 k=1 2. [|q| < 1] ka 1 = bk (b − 1)a+1 [|q| < 1] (cf. 113) (cf. 511) a ∞ k=1 π 1 = (4k − 1)(4k + 1) 4 ⎡ ⎣ 1 ba−i i=1 i j=0 (cf. (a + 1 − j)! [a = 1, 2, 3, . . 233 ∞ 1. k=1 1 1 1 = 1 + p + p + . . = ζ(p) kp 2 3 ∞ (−1)k+1 2. 10 k=1 1 22n−1 π 2n |B2n |, = k 2n (2n)! ∞ (−1)k+1 4. k=1 ∞ 5. k=1 (−1)k+1 k=1 ∞ k=1 1 π2 = k2 6 1 (22n−1 − 1)π 2n |B2n | = k 2n (2n)! 1 (22n − 1)π 2n |B2n | = 2n (2k − 1) 2 · (2n)! ∞ 6. 1 = (1 − 21−p ) ζ(p) kp 1 π 2n+1 |E2n | = 2n+2 2n+1 (2k − 1) 2 (2n)!
K=n+1 n (−1)r 2. 211 uk = u1 + u2 + u3 + . . 212 k=1 composed of the absolute values of its terms converges. 211 is said to converge conditionally. Every absolutely convergent series converges. 211 converges absolutely. 211 diverges. 211 converges absolutely. 211 diverges. 211 diverges. 211 converges absolutely. 211 diverges. 224 Suppose that f (x) is a positive decreasing function and that ek f ek lim =q k→∞ f (k) ∞ for natural k. If q < 1, the series k=1 f (k) converges. If q > 1, this series diverges.
Table of integrals, series and products by I S Gradshteĭn; I M Ryzhik; Alan Jeffrey; Daniel Zwillinger
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