#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Products ---------- " User User #4: " => [640 pi^3]/[2187 sqrt(3)] [Gradshteyn and Ryzhik 8.338(5)] " 8 / k \ #5: PRODUCT GAMMA|---| User k=1 \ 3 / 3 640*SQRT(3)*pi #6: ----------------- Simp(#5) 6561 #7: " => n! = gamma(n + 1) " User #8: n :epsilon Integer User #9: n Simp(#8) n #10: PRODUCT k User k=1 #11: n! Simp(#10) #12: " => x^[n (n + 1)/2] " User n k #13: PRODUCT x User k=1 n*(n + 1)/2 #14: x Simp(#13) #15: " => n " User n - 1 / 1 \ #16: PRODUCT |1 + ---| User k=1 \ k / #17: n Simp(#16) #18: " => 1/2^(2 n) binomial(2 n, n) [Knopp, p. 385] " User n 2*k - 1 #19: PRODUCT --------- User k=1 2*k / 1 \ |n - ---|! #20: \ 2 / Simp(#19) ------------- SQRT(pi)*n! User #21: " => [x^(2 n) - 1]/(x^2 - 1) [Gradshteyn and Ryzhik 1.396(1)] " n - 1 / 2 / k*pi \ \ #22: PRODUCT |x - 2*x*COS|------| + 1| User k=1 \ \ n / / n - 1 / / pi*k \ 2 \ #23: PRODUCT |- 2*x*COS|------| + x + 1| Simp(#22) k=1 \ \ n / / #24: " => 2/3 [Knopp, p. 228] " User 3 inf k - 1 #25: PRODUCT -------- User k=2 3 k + 1 / inf 3 \ inf 1 |PRODUCT (k - 1)|*PRODUCT -------- #26: \ k=2 / k=2 3 Simp(#25) k + 1 #27: " => 2/pi [Gradshteyn and Ryzhik 0.262(2)] " User inf / 1 \ PRODUCT |1 - --------| #28: k=1 | 2 | User \ (2*k) / 2 #29: ---- Simp(#28) pi #30: " => sqrt(2) [Gradshteyn and Ryzhik 0.261] " User / k + 1 \ inf | (-1) | #31: PRODUCT |1 + -----------| User k=1 \ 2*k - 1 / inf / COS(pi*k) \ #32: PRODUCT |----------- + 1| Simp(#31) k=1 \ 1 - 2*k / #33: " => -1 [Knopp, p. 436] " User inf k*(k + 1) + 1 + #i #34: PRODUCT -------------------- User k=0 k*(k + 1) + 1 - #i Simp(#34) / 3 2 inf | k*(k + 2*k + 3*k + 2) #35: PRODUCT |---------------------------- + k=0 | 4 3 2 \ k + 2*k + 3*k + 2*k + 2 2 \ 2*#i*(k + k + 1) | ----------------------------| 4 3 2 | k + 2*k + 3*k + 2*k + 2 / #36: " ---------- Quit ---------- " User