#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Sums ---------- " User User #4: " Simplify the sum below to sum(x[i]^2, i = 1..n) - sum(x[i], i = 1..n)^2/n " #5: n :epsilon Integer User #6: n Simp(#5) n SUM x #7: j=1 j User xbar := -------- n n 2 #8: SUM (x - xbar) User i=1 i / n \ n / n \2 2*|SUM x |*SUM x |SUM x | #9: / n 2\ \i=1 i/ j=1 j \j=1 j/ Simp(#8) |SUM x | - ------------------- + ----------- \i=1 i / n n / n \ n / n \2 2*|SUM x |*SUM x |SUM x | #10: / n 2\ \i=1 i/ i=1 i \i=1 i/ Sub(#9) |SUM x | - ------------------- + ----------- \i=1 i / n n / n \2 |SUM x | #11: / n 2\ \i=1 i/ Simp(#10) |SUM x | - ----------- \i=1 i / n #12: xbar := User User #13: " Derivation of the least squares fitting of data points (x[i], y[i]) to a " User #14: " line y = m x + b. See G. Keady, ``Using Maple's linalg package with Zill " User #15: " and Cullen _Advanced Engineering Mathematics_, Part II: Vectors, Matrices " User #16: " and Vector Calculus'', University of Western Australia, " #17: " ftp://maths.uwa.edu.au/pub/keady/ " User n 2 #18: f := SUM (y - m*x - b) User i=1 i i //d d \ \ #19: SOLVE||-- f = 0, -- f = 0|, [m, b]| User \\dm db / / Simp(#19) / n / n \ n | n*SUM x *y - |SUM x |*SUM y | i=1 i i \i=1 i/ i=1 i #20: | m = ------------------------------- b = | n 2 / n \2 | n*SUM x - |SUM x | \ i=1 i \i=1 i/ / n 2\ n / n \ n \ |SUM x |*SUM y - |SUM x |*SUM x *y | \i=1 i / i=1 i \i=1 i/ i=1 i i | --------------------------------------- | n 2 / n \2 | n*SUM x - |SUM x | | i=1 i \i=1 i/ / #21: f := User User #22: " Indefinite sum => (-1)^n binomial(2 n, n). See Herbert S, Wilf, " User #23: " ``IDENTITIES and their computer proofs'', University of Pennsylvania. " k 2 #24: SUM (-1) *COMB(2*n, k) User k Simp(#24) 2 COS(pi*k) 2 SIN(pi*k) (2*n)! *SUM ----------------- + #i*(2*n)! *SUM ----------------- #25: k 2 2 k 2 2 k! *(2*n - k)! k! *(2*n - k)! User #26: " Check whether the full Gosper algorithm is implemented " #27: " => 1/2^(n + 1) binomial(n, k - 1) " User / COMB(n, k) COMB(n + 1, k) \ SUM |------------ - ----------------| #28: k | n n + 1 | User \ 2 2 / Simp(#28) -n - 1 / 1 #29: - 2 *n!*|(n + 1)*SUM ------------------------------------ + \ k (k - n - 1)*(k - 1)!*(-k + n + 1)! 1 / 2*SUM ------------------------ - (n + 1)*|(n + 1)*SUM k (k - 1)!*(-k + n + 1)! \ k 1 1 \\ ------------------------------ + 2*SUM ------------------|| (k - n - 1)*k!*(-k + n + 1)! k k!*(-k + n + 1)! // User #30: " Dixon's identity (check whether Zeilberger's algorithm is implemented). " User #31: " Note that the indefinite sum is equivalent to the definite " User #32: " sum(..., k = -min(a, b, c)..min(a, b, c)) => (a + b + c)!/(a! b! c!) " #33: " [Wilf] " User User k #34: SUM (-1) *COMB(a + b, a + k)*COMB(b + c, b + k)*COMB(c + a, c + k k) Simp(#34) #35: (a + b)!*(a + c)!*(b + c)!*SUM k COS(pi*k) ------------------------------------------------------- + (a - k)!*(b - k)!*(c - k)!*(k + a)!*(k + b)!*(k + c)! #i*(a + b)!*(a + c)!*(b + c)!*SUM k SIN(pi*k) ------------------------------------------------------- (a - k)!*(b - k)!*(c - k)!*(k + a)!*(k + b)!*(k + c)! #36: " Telescoping sum => g(n + 1) - g(0) " User #37: G(k) := User n #38: SUM (G(k + 1) - G(k)) User k=0 #39: G(n + 1) - G(0) Simp(#38) #40: g := User #41: " => n^2 (n + 1)^2 / 4 " User n 3 #42: SUM k User k=1 2 2 n *(n + 1) #43: ------------- Simp(#42) 4 User #44: " See Daniel I. A. Cohen, _Basic Techniques of Combinatorial Theory_, John " User #45: " Wiley and Sons, 1978, p. 60. The following two sums can be derived directly " #46: " from the binomial theorem: " User User #47: " sum(k^2 * binomial(n, k) * x^k, k = 1..n) = n x (1 + n x) (1 + x)^(n - 2) " #48: " => n (n + 1) 2^(n - 2) [Cohen, p. 60] " User n 2 #49: SUM k *COMB(n, k) User k=1 2 n k #50: n!*SUM ------------- Simp(#49) k=1 k!*(n - k)! #51: " => [2^(n + 1) - 1]/(n + 1) [Cohen, p. 83] " User n COMB(n, k) #52: SUM ------------ User k=0 k + 1 n 1 #53: n!*SUM ------------------- Simp(#52) k=0 (k + 1)!*(n - k)! User #54: " Vandermonde's identity => binomial(n + m, r) [Cohen, p. 31] " r #55: SUM COMB(n, k)*COMB(m, r - k) User k=0 r 1 #56: m!*n!*SUM ----------------------------------- Simp(#55) k=0 k!*(k + m - r)!*(n - k)!*(r - k)! #57: " => Fibonacci[2 n] [Cohen, p. 88] " User n #58: SUM COMB(n, k)*FIBONACCI(k) User k=0 // 0 1 \k \ || | . [0, 1]| \\ 1 1 / / #59: n 1 Simp(#58) n!*SUM ----------------------- k=0 k!*(n - k)! #60: " => Fibonacci[n] Fibonacci[n + 1] [Cohen, p. 65] " User n 2 #61: SUM FIBONACCI(k) User k=1 n // 0 1 \k \ 2 SUM || | . [0, 1]| #62: k=1 \\ 1 1 / / Simp(#61) 1 #63: " => 1/2 cot(x/2) - cos([2 n + 1] x/2)/[2 sin(x/2)] " User User #64: " See Konrad Knopp, _Theory and Application of Infinite Series_, Dover " #65: " Publications, Inc., 1990, p. 480. " User n #66: SUM SIN(k*x) User k=1 / x \ / / 1 \\ COT|---| COS|x*|n + ---|| \ 2 / \ \ 2 // #67: ---------- - ------------------ Simp(#66) 2 / x \ 2*SIN|---| \ 2 / User #68: " => sin(n x)^2/sin x [Gradshteyn and Ryzhik 1.342(3)]" n #69: SUM SIN((2*k - 1)*x) User k=1 1 COS(2*n*x) #70: ---------- - ------------ Simp(#69) 2*SIN(x) 2*SIN(x) #71: " Simplify the previous expression! " User #72: Trigonometry := Expand User 2 SIN(n*x) #73: ----------- Simp(#70) SIN(x) #74: Trigonometry := Auto User #75: " => Fibonacci[n + 1] [Cohen, p. 87] " User FLOOR(n/2) #76: SUM COMB(n - k, k) User k=0 FLOOR(n/2) (n - k)! #77: SUM --------------- Simp(#76) k=0 k!*(n - 2*k)! #78: " => pi^2 / 6 + zeta(3) =~ 2.84699 " User inf / 1 1 \ q_ := SUM |---- + ----| #79: k=1 | 2 3 | User \ k k / #80: q_ User 2 pi #81: ZETA(3) + ----- Simp(#80) 6 #82: Precision := Approximate User #83: APPROX(q_) User #84: 2.84699 Simp(#83) #85: Precision := Exact User User #86: " => pi^2/12 - 1/2 (log 2)^2 [Gradshteyn and Ryzhik 0.241(2)] " inf 1 SUM ------- #87: k=1 k 2 User 2 *k -k inf 2 #88: SUM ----- Simp(#87) k=1 2 k #89: " => pi/12 sqrt(3) - 1/4 log 3 [Knopp, p. 268] " User inf 1 #90: SUM ------------------------------- User k=0 (3*k + 1)*(3*k + 2)*(3*k + 3) SQRT(3)*pi LN(3) #91: ------------ - ------- Simp(#90) 12 4 User #92: " => 1/2 (2^(n - 1) + 2^(n/2) cos(n pi/4)) [Gradshteyn and Ryzhik 0.153(1)] " inf #93: SUM COMB(n, 4*k) User k=0 inf 1 #94: n!*SUM ------------------- Simp(#93) k=0 (n - 4*k)!*(4*k)! #95: " => 1 [Knopp, p. 233] " User inf 1 #96: SUM ----------------------------------------- User k=1 SQRT(k*(k + 1))*(SQRT(k) + SQRT(k + 1)) / / / 1 1 \ / 1 1 \\\ #97: | lim |ZETA|---, --- + 1| - ZETA|---, ---||| + 1 Simp(#96) \k->0+ \ \ 2 k / \ 2 k /// User #98: " => 1/sqrt([1 - x y]^2 - 4 x^2) (| x y | < 1 and -1 <= x < 1). " User #99: " From Evangelos A. Coutsias, Michael J. Wester and Alan S. Perelson, ``A " User #100:" Nucleation Theory of Cell Surface Capping'', draft. " User inf FLOOR(n/2) n n - 2*k #101:SUM SUM COMB(n, k)*COMB(n - k, n - 2*k)*x *y n=0 k=0 - 2*k inf n n FLOOR(n/2) y #102:SUM x *y *n!* SUM ---------------- Simp(#101) n=0 k=0 2 k! *(n - 2*k)! #103:" An equivalent summation to the above is: " User inf inf n! / x \k n - k SUM SUM ----------------*|---| *(x*y) #104:k=0 n=2*k 2 \ y / User k! *(n - 2*k)! n - 2*k inf n!*(x*y) |y| * SUM ------------ #105:inf n=2*k (n - 2*k)! Simp(#104) SUM ----------------------------- k=0 2 k! #106:" => pi/2 [Knopp, p. 269] " User inf m k #107:SUM PRODUCT --------- User m=2 k=1 2*k - 1 -m inf 2 *m! SQRT(pi)*SUM ------------ #108: m=2 / 1 \ Simp(#107) |m - ---|! \ 2 / #109:" ---------- Quit ---------- " User