1kill (all); 2done; 3 4(assume_pos_save: assume_pos, assume_pos: true); 5true; 6 7/* Known failure: concat (q, k) => gensym 8 */ 9sum (concat (q, k), k, a, b); 10(b - a + 1) * qk; 11 12/* Known failure: concat (q, k) => gensym 13 */ 14product (concat (q, k), k, a, b); 15qk ^ (b - a + 1); 16 17/* Known failure: causes asksign to ask an unanswerable question about index variable; 18 * this is a bug in integrate and/or asksign. 19(assume (b > 0, a > 0, b > a), sum (integrate (1/x^k, x, a, b), k, 1, n)); 20integrate (1/x, x, a, b) + sum (integrate (1/x^k, x, a, b), k, 2, n); 21 */ 22 23/* Known failure: causes asksign to ask an unanswerable question about index variable; 24 * this is a bug in integrate and/or asksign. 25product (integrate (1/x^k, x, a, b), k, 1, n); 26integrate (1/x, x, a, b) * product (integrate (1/x^k, x, a, b), k, 2, n); 27 */ 28 29block ([prederror: false], sum(if k < 1 then a else b,k,0,n)); 30'( 'sum(if k < 1 then a else b,k,0,n)); 31 32/* see SF bug report # 625278 */ 33 34'sum(binomial(2,2-k)-binomial(2,1-k),k,1,2),simpsum; 352; 36 37'sum(binomial(2,2-k)-binomial(2,1-k),k,1,2),sum; 382; 39 40'sum(binomial(x,2-k)-binomial(x,1-k),k,1,2),simpsum; 41x; 42 43'sum(binomial(x,2-k)-binomial(x,1-k),k,1,2),sum; 44x; 45 46sum (f(k) + 1, k, 1, n), simpsum; 47n + 'sum (f(k), k, 1, n); 48 49/* further examples in the same vein: part of summand depends on index, part doesn't. */ 50 51sum (sin(g(j)) - %pi^j + cos(h(k)) - %e^l, j, 1, m), simpsum; 52m*cos(h(k)) - m*%e^l + sum (sin(g(j)), j, 1, m) - ((%pi^(m + 1) - %pi)/(%pi - 1)); 53 54sum (sum (f(j) + g(k), j, 1, m), k, 1, n), simpsum; 55m * 'sum (g(k), k, 1, n) + n * 'sum (f(j), j, 1, m); 56 57sum (sum (f(j) + g(k) - %pi^l + 2, j, 1, m), k, 1, n), simpsum; 58m * 'sum (g(k), k, 1, n) + n * 'sum (f(j), j, 1, m) - m*n*%pi^l + 2*m*n; 59 60sum (sum (sum (%pi + %phi - 1, i, 1, n_i), j, 1, n_j), k, 1, n_k); 61n_i * n_j * n_k * (%pi + %phi - 1); 62 63sum (sum (sum (f(%pi, i) + g(%phi, j) - h(k), i, 1, n_i), j, 1, n_j), k, 1, n_k); 64'sum ('sum ('sum (f(%pi, i) + g(%phi, j) - h(k), i, 1, n_i), j, 1, n_j), k, 1, n_k); 65 66/* see SF bug report # 649428 */ 67 68(factsum3 (mt, ej) := sum ((-1)^(k + 1)/(k*mt^k)*sum((1 - l)^k - l^k, l, 1, ej),k,1,inf), t% : taylor (factsum3 (mt, ej), [mt, 0, 3, asymp])); 69'sum(1-2*l,l,1,ej)/mt-('sum(1-2*l,l,1,ej)/(2*mt^2))+'sum(-2*l^3+3*l^2-3*l+1,l,1,ej)/(3*mt^3); 70 71%, simpsum, ratsimp; 72-((6*ej^2*mt^2-3*ej^2*mt+ej^4+ej^2)/(6*mt^3)); 73 74/* Known failure: interaction of taylor and simpsum 75 */ 76taylor (factsum3 (mt, ej), [mt, 0, 3, asymp]), simpsum; 77-(6*ej^2*mt^2-3*ej^2*mt+ej^4+ej^2)/(6*mt^3); 78 79/* see SF bug report # 740134 */ 80 81(foo: i^2, [sum ('foo, i, 0, 2), sum (foo, i, 0, 2), sum (foo, i, 0, n)]); 82[3*foo, 5, 'sum (i^2, i, 0, n)]; 83 84/* summand is a function which has a side effect -- 85 * outcome here differs from recommendation by Stavros in 740134 86 */ 87block ([L:[]], f(i) := (L: append (L, [i]), i), S: sum (f (i), i, 1, 3), [S, L]); 88[6, [1, 2, 3]]; 89 90sum (integrate (x^i ,x), i, 0, 2); 91x^3/3 + x^2/2 + x; 92 93sum (integrate (1/(x^i + 1), x), i, 0, 1); 94log(x+1) + x/2; 95 96(f[i](x) := x^i, g[i](x) := x^i, h[i](x) := x^i, 0); 970; 98 99(f[i], g[i](t), 0); 1000; 101 102sum (f[i](x), i, 0, n); 103'sum (x^i, i, 0, n); 104 105sum (g[i](x), i, 0, n); 106'sum (x^i, i, 0, n); 107 108sum (h[i](x), i, 0, n); 109'sum (x^i, i, 0, n); 110 111/* see SF bug report # 817521 */ 112 113(kill (foo), foo (n):= block([aaa,m], modedeclare([m,n], fixnum), m:n, [ aaa[1], aaa[n], aaa[m], aaa[1]+aaa[2], sum(aaa[i],i,1,m) ] ), foo (2)); 114[aaa[1],aaa[2],aaa[2],aaa[2]+aaa[1], aaa[2]+aaa[1]]; 115 116(translate(foo), 0); 1170; 118 119foo (2); 120[aaa[1],aaa[2],aaa[2],aaa[2]+aaa[1], aaa[2]+aaa[1]]; 121 122/* see SF bug report # 851765 */ 123 124(apply (forget, facts ()), assume (i >= a), sum (i, i, a, b), facts ()); 125[i >= a]; 126 127is (i >= a); 128true; 129 130(assume (i < 0), sum (abs (i), i, 1, a)); 131'sum (i, i, 1, a); 132 133(apply (forget, facts ()), 0); 1340; 135 136/* see SF bug report # 1007094 */ 137 138(u: 'sum((-1/(8*i+6)-1/(8*i+5)-2/(8*i+4)+4/ (8*i+1))/16^i,i,1,a), 0); 1390; 140 141u, a=2, simpsum; 1421618091/196035840; 143 144%, numer; 1450.0082540570132481894$ 146 147u, a=2, simpsum, numer; 1480.0082540570132481911$ 149 150u, a=2, simpsum, bfloat; 1518.25405701324819B-3; 152 153/* see SF bug report # 1192935 */ 154 155(f(x) := x^2, b: 1/10, Ak: b*f(k*b), sum (Ak, k, 0, 9)); 15657/200; 157 158/* When A(k) is defined as A(k) := b*f(k*b) instead of define(A(k), b*f(k*b)), 159 * A(k) evaluates to k^2/n^3 even if n is bound to something; 160 * this is a consequence of Maxima's evaluation policies: 161 * rhs of := is not evaluated, and variables (b in this case) 162 * are evaluated just once. 163 */ 164(b:1/n, define (A(k), b*f(k*b)), A(k)); 165k^2/n^3; 166 167sum (A(k), k, 0, n-1); 168('sum (k^2, k, 0, n-1))/n^3; 169 170sum (A(k), k, 0, n-1), simpsum; 171(n+2*(n-1)^3+3*(n-1)^2-1)/(6*n^3); 172 173[''%, sum (A(k), k, 0, n-1)], n=10; 174[57/200, 57/200]; 175 176(B(k) := k*f(k*b), B(k)); 177k^3/n^2; 178 179sum (B(k), k, 0, n-1), simpsum; 180((n-1)^4 + 2*(n-1)^3 + (n-1)^2) / (4*n^2); 181 182/* see SF bug report # 1363411 */ 183 184(kill (f), 'sum(1+f(k),k,1,2)), simpsum; 1852 + f(1) + f(2); 186 187is(equal(block([gcd:subres], ratsimp((-x^3+1)/(2*x^5+2) + (-sqrt(5)+5)/(20*x^2+(-10*sqrt(5)-10)*x+20))), 188 block([gcd:spmod], ratsimp((-x^3+1)/(2*x^5+2) + (-sqrt(5)+5)/(20*x^2+(-10*sqrt(5)-10)*x+20))))); 189true; 190 191/* General tests from here on */ 192 193sum (x, x, p, p + 2); 194p + (p + 1) + (p + 2); 195 196product (x, x, p, p + 2); 197p * (p + 1) * (p + 2); 198 199sum (x, x, %i, %i + 2); 200%i + (%i + 1) + (%i + 2); 201 202product (x, x, %i, %i + 2); 203%i * (%i + 1) * (%i + 2); 204 205sum (imagpart (1 + %i*k), k, 1, n), simpsum; 206(n^2 + n)/2; 207 208product (imagpart (1 + %i*k), k, 1, n), simpproduct; 209n!; 210 211(g(x) := q(x^2), sum (g(k), k, 1, n)); 212'sum (q (k^2), k, 1, n); 213 214(g(x) := q(x^2), product (g(k), k, 1, n)); 215'product (q (k^2), k, 1, n); 216 217sum (integrate (x^k, x, 0, 1), k, 1, n); 218'sum (1/(k + 1), k, 1, n); 219 220product (integrate (x^k, x, 0, 1), k, 1, n); 221'product (1/(k + 1), k, 1, n); 222 223(kill(b), sum ('concat (q, k), k, a, b)); 224'sum ('concat (q, k), k, a, b); 225 226sum ('concat (q, k), k, a, b), nouns, a=1, b=5; 227q5 + q4 + q3 + q2 + q1; 228 229product ('concat (q, k), k, a, b); 230'product ('concat (q, k), k, a, b); 231 232product ('concat (q, k), k, a, b), nouns, a=1, b=5; 233q5 * q4 * q3 * q2 * q1; 234 235(assume (n>= 1), sum (if k <= n then a else b, k, 1, n)); 236a * n$ 237 238product (if k <= n then a else b, k, 1, n); 239a ^ n$ 240 241sum (sin (%pi * k), k, 0, nn); 2420$ 243 244product (cos (2 * %pi * k), k, 0, nn); 2451$ 246 247sum (lambda ([x], x^i), i, 1, 3); 248lambda ([x], x) + lambda ([x], x^2) + lambda ([x], x^3); 249 250sum (k: k^2, k, 1, 3); 25114; 252 253sum (a, k, inf, minf); 2540; 255 256(assume (i < 0, j < 0, n > 0), sum(sum(abs(i) + abs(j),i,1,j),j,1,n)); 257'sum ('sum (i + j, i, 1, j), j, 1, n); 258 259sum (sum (sqrt (i + j), i, 1, j), j, 1, n), simpsum; 260'sum ('sum (sqrt (i + j), i, 1, j), j, 1, n); 261 262sum (1/k, k, rat(1), rat(n)); 263'sum (1/k, k, 1, n); 264 265sum (taylor (x^i, x, 0, 5), i, 0, 5); 266''(taylor (x^5 + x^4 + x^3 + x^2 + x + 1,x,0,5)); 267 268(i: 888, sum (lambda ([i], i^2), i, 1, 3)); 2693 * lambda ([i], i^2); 270 271sum (lambda ([i], i^2), i, 1, n); 272n * lambda ([i], i^2); 273 274(f(x) := sum (x, i, 1, 3), f(-x)); 275-3*x; 276 277(ak : k^2, g(a,n) := sum(a,k,1,n), g(ak, 5)); 27855; 279 280(translate (g), g(ak,5)); 28155; 282 283sum (integrate (x^i, x), i, 0, n); 284'sum (x^(i+1) / (i+1), i, 0, n); 285 286(assume (k < 0), assume (0 < n, m > n), sum (sqrt (k^2), k, n, m)); 287'sum (k, k, n, m); 288 289(assume (i >= n_1, i <= n_2), sum (z(i), i, n_1, n_2)); 290'sum (z(i), i, n_1, n_2); 291 292block ([prederror: false], [is (i >= n_1), is (i <= n_2)]); 293[true, true]; 294 295/* some examples adapted from describe("sum") */ 296 297(old_facts: facts (), foo: 42, bar: 12 + exp(7), baz: 1/quux, 0); 2980; 299 300sum (foo^2, foo, 1, 7); 301140; 302 303sum (aa[bar], bar, 1, 7); 304aa[7] + aa[6] + aa[5] + aa[4] + aa[3] + aa[2] + aa[1]; 305 306sum (aa(baz), baz, 1, 7); 307aa(7) + aa(6) + aa(5) + aa(4) + aa(3) + aa(2) + aa(1); 308 309sum (aa(foo), foo, 1, n); 310'sum (aa(foo), foo, 1, n); 311 312sum (2^bar + bar^2, bar, 0, n), simpsum; 3132^(n + 1) + (2*n^3 + 3*n^2 + n)/6 - 1; 314 315sum (1/3^baz, baz, 1, inf), simpsum; 3161/2; 317 318sum (foo^2, foo, 1, 4) * sum (1/foo^2, foo, 1, inf), simpsum; 3195*%pi^2; 320 321(gg(bar) := 1/bar^2, sum (gg(bar)^2, bar, 1, inf)); 322sum (1/bar^4, bar, 1, inf); 323 324sum (gg(baz)^2, baz, 1, inf), simpsum; 325%pi^4/90; 326 327[is (foo = 42), is (bar = 12 + exp(7)), is (baz = 1/quux)]; 328[true, true, true]; 329 330(new_facts: facts(), is (sort (old_facts) = sort (new_facts))); 331true; 332 333/* redo all of the tests, this time with product */ 334 335product (lambda ([x], x^i), i, 1, 3); 336lambda ([x], x) * lambda ([x], x^2) * lambda ([x], x^3); 337 338product (k: k^2, k, 1, 3); 33936; 340 341product (a, k, inf, minf); 3421; 343 344(assume (i < 0, j < 0, n > 0), product(product(abs(i) + abs(j),i,1,j),j,1,n)); 345'product ('product (i + j, i, 1, j), j, 1, n); 346 347product (1/k, k, rat(1), rat(n)); 348'product (1/k, k, 1, n); 349 350product (taylor (x^i, x, 0, 5), i, 0, 5); 351''(taylor (x^15,x,0,15)); 352 353(i: 888, product (lambda ([i], i^2), i, 1, 3)); 354(lambda ([i], i^2))^3; 355 356product (lambda ([i], i^2), i, 1, n); 357lambda([i],i^2)^n; 358 359(f(x) := product (x, i, 1, 3), f(-x)); 360-x^3; 361 362(f(x) := x^2, b: 1/10, Ak: b*f(k*b), product (Ak, k, 1, 9)); 363(9!)^2/1000^9; 364 365(A(k) := k*f(k*b), b:1/n, A(k)); 366k^3/n^2; 367 368product (A(k), k, 0, n-1); 369('product (k^3/n^2, k, 0, n-1)); 370 371(assume (n > 0, m > n), ev (product (k, k, n, m), simpproduct)); 372m!/(n - 1)!; 373 374(ak : k^2, g(a,n) := product(a,k,1,n), g(ak, 5)); 375(5!)^2; 376 377(translate (g), g(ak,5)); 378(5!)^2; 379 380/* multiplicand is a function which has a side effect */ 381 382block ([L:[]], f(i) := (L: append (L, [i]), i), S: product (f (i), i, 1, 3), [S, L]); 383[6, [1, 2, 3]]; 384 385product (integrate (x^i ,x), i, 0, 2); 386x^3/3 * x^2/2 * x; 387 388product (integrate (1/(x^i + 1), x), i, 0, 1); 389log(x+1) * x/2; 390 391(f[i](x) := x^i, g[i](x) := x^i, h[i](x) := x^i, 0); 3920; 393 394/* reference f[i] and g[i] -- see 740134 for the effect this has on previous defn of sum */ 395 396(f[i], g[i](t), 0); 3970; 398 399product (f[i](x), i, 0, n); 400'product (x^i, i, 0, n); 401 402product (g[i](x), i, 0, n); 403'product (x^i, i, 0, n); 404 405product (h[i](x), i, 0, n); 406'product (x^i, i, 0, n); 407 408product (integrate (x^i, x), i, 0, n); 409'product (x^(i+1) / (i+1), i, 0, n); 410 411(assume (k < 0), assume (0 < n, m > n), ev (product (sqrt (k^2), k, n, m), simpproduct)); 412m!/(n - 1)!; 413 414(assume (i >= n_1, i <= n_2), product (z(i), i, n_1, n_2)); 415'product (z(i), i, n_1, n_2); 416 417block ([prederror: false], [is (i >= n_1), is (i <= n_2)]); 418[true, true]; 419 420/* some examples adapted from describe("sum") */ 421 422(old_facts: facts (), foo: 42, bar: 12 + exp(7), baz: 1/quux, 0); 4230; 424 425product (foo^2, foo, 1, 7); 426(7!)^2; 427 428product (aa[bar], bar, 1, 7); 429aa[7] * aa[6] * aa[5] * aa[4] * aa[3] * aa[2] * aa[1]; 430 431product (aa(baz), baz, 1, 7); 432aa(7) * aa(6) * aa(5) * aa(4) * aa(3) * aa(2) * aa(1); 433 434product (aa(foo), foo, 1, n); 435product (aa(foo), foo, 1, n); 436 437product (2^bar + bar^2, bar, 0, n), simpsum; 438product (2^bar + bar^2, bar, 0, n); 439 440product (1/3^baz, baz, 1, inf), simpsum; 441product (1/3^baz, baz, 1, inf); 442 443product (foo^2, foo, 1, 4) * product (1/foo^2, foo, 1, inf), simpsum; 444(4!)^2 * 'product (1/'foo^2, 'foo, 1, inf); 445 446(gg(bar) := 1/bar^2, product (gg(bar)^2, bar, 1, inf)); 447'product (1/'bar^4, 'bar, 1, inf); 448 449product (gg(baz)^2, baz, 1, inf), simpsum; 450'product (1/'baz^4, 'baz, 1, inf); 451 452[is (foo = 42), is (bar = 12 + exp(7)), is (baz = 1/quux)]; 453[true, true, true]; 454 455(new_facts: facts(), is (sort (old_facts) = sort (new_facts))); 456true; 457 458/*---- new tests from bw ------*/ 459 460sum(0,i,a,b); 4610$ 462 463sum(0,i,minf,inf); 4640$ 465 466sum(0,i,42,inf); 4670$ 468 469sum(sqrt(3),i,0,inf); 470inf$ 471 472sum(sqrt(3),i,-1932,inf); 473inf$ 474 475sum(5,i,minf, inf); 476inf$ 477 478sum(-sqrt(3),i,0,inf); 479minf$ 480 481sum(sqrt(3),i,1,3); 4823 * sqrt(3)$ 483 484sum(sqrt(3),i,-1,1); 4853 * sqrt(3)$ 486 487sum(p(i + 1) - p(i),i,1,5); 488p(6) - p(1)$ 489 490product(1,i,a,b); 4911$ 492 493product(1,i,minf,inf); 4941$ 495 496product(1,i,-42,inf); 4971$ 498 499product(%pi,i,1,3); 500%pi^3$ 501 502product(%pi,i,-1,1); 503%pi^3$ 504 505product(%pi,i,0,inf); 506inf$ 507 508product(%pi - 3,i,0,inf); 5090$ 510 511product(%pi - 3,i,minf,inf); 5120$ 513 514product(-%pi,i,0,inf); 515infinity$ 516 517product(0,i,0,n); 5180$ 519 520product(-%pi,i,minf,inf); 521infinity$ 522 523product((1 + %i)/sqrt(2),k,1,inf); 524und$ 525 526product(-1,k,1,inf); 527und$ 528 529product(a,i,0,inf); 530'product(a,i,0,inf); 531 532product(p(i + 1) / p(i),i,1,5); 533p(6) / p(1)$ 534 535sum(lambda([x],x^i),i,1,3); 536lambda([x],x) + lambda([x],x^2) + lambda([x],x^3); 537 538product(lambda([x],x^i),i,1,3); 539lambda([x],x) * lambda([x],x^2) * lambda([x],x^3); 540 541sum([1,k],k,1,5); 542[5,15]; 543 544sum([1,k],k,1,n),simpsum; 545[n, (n^2 + n)/2]; 546 547(old_facts : facts(),0); 5480$ 549 550errcatch(sum(1/k,k,-1,1)); 551[]; 552 553(new_facts: facts(), is (sort (old_facts) = sort (new_facts))); 554true; 555 556(old_facts : facts(),0); 5570$ 558 559errcatch(product(1/k,k,-1,1)); 560[]; 561 562(new_facts: facts(), is (sort (old_facts) = sort (new_facts))); 563true; 564 565sum(diff(log(x),x,k),k,1,2); 5661/x - 1/x^2$ 567 568product(diff(log(x),x,k),k,1,2); 569-1/x^3; 570 571sum(x^k * at(diff(sin(x),x,k)/k!,x=0),k,0,5) - taylor(sin(x),x,0,5); 572''(taylor(0,x,0,5)); 573 574product(%i-x,k,1,inf); 575'product(%i-x,k,1,inf); 576 577sum(%i,k,0,-1); 5780$ 579 580sum(%i,k,inf,minf); 5810$ 582 583sum(%i,k, 0, minf); 5840$ 585 586product(%i,k,0,-1); 5871$ 588 589product(%i,k,inf,minf); 5901$ 591 592product(%i,k, 0, minf); 5931$ 594 595sum(p,p,p,p); 596p$ 597 598product(p,p,p,p); 599p$ 600 601sum(k,k,p,p+1); 6022*p+1$ 603 604sum(k,k,p,p-1); 6050$ 606 607sum(k,k,%i,%i+1); 6082*%i + 1$ 609 610product(k,k,%i,%i+1); 611%i * (%i + 1)$ 612 613product(k,k,p,p+1); 614p*(p+1)$ 615 616product(k,k,p,p-1); 6171$ 618 619sum(k,k,1,entier(4/3)); 6201$ 621 622sum(k,k,entier(4/3),1); 6231$ 624 625sum(a,k,entier(4/3),entier(5/2)); 6262*a$ 627 628product(k,k,1,entier(4/3)); 6291$ 630 631product(k,k,entier(4/3),1); 6321$ 633 634product(a,k,entier(4/3),entier(5/2)); 635a^2$ 636 637sum(q(k),k,n^2 + 1,-n^2 - 1); /* It's an empty sum */ 6380$ 639 640product(q(k),k,n^2 + 1,-n^2 - 1); 6411$ 642 643sum(concat(q,k),k,1,3); 644q1 + q2 + q3$ 645 646(s : sum('concat(q,k),k,1,n),0); 6470$ 648 649(s : subst(n=3,s), ev(s,nouns)); 650q1 + q2 + q3$ 651 652sum(product(concat(q,i),i,1,j),j,1,3); 653q1 + q1 * q2 + q1 * q2 * q3$ 654 655product(concat(q,k),k,1,3); 656q1 * q2 * q3$ 657 658(s : product('concat(q,k),k,1,n),0); 6590$ 660 661(s : subst(n=3,s), ev(s,nouns)); 662q1 * q2 * q3$ 663 664(a : k^2,0); 6650$ 666 667sum(realpart(a + %i * k),k,1,3); 66814$ 669 670(assume(k < 0),0); 6710$ 672 673sum(sum(abs(k),k,1,n),n,1,m); 674sum(sum(k,k,1,n),n,1,m)$ 675 676sum(sum(k,k,1,i),i,1,n),simpsum,ratsimp; 677''(ratsimp(n^3/6+n^2/2+n/3))$ 678 679sum(sum(abs(k),k,1,n) + sum(abs(k),k,1,n),n,1,m); 6802 * sum(sum(k,k,1,n),n,1,m)$ 681 682(forget(k < 0),0); 6830$ 684 685(remvalue(a,b),0); 6860$ 687 688sum(if k < 1 then a else b,k,0,2); 689a + 2*b$ 690 691block ([prederror : true], sum(if k < 1 then a else b,k,0,2)); 692a + 2*b$ 693 694sum(1/3^k,k,1,inf),simpsum; 6951/2$ 696 697sum(3^-i,i,1,inf),simpsum; 6981/2$ 699 700(kill(x), assume(abs(x)-1 < 0)); 701[abs(x) < 1]; 702 703sum(x^k,k,0,inf),simpsum; 7041/(1-x)$ 705 706(forget(abs(x) - 1 < 0),0); 7070$ 708 709/*--- A few challenging sums ----*/ 710 711(assume (n >= 1), sum (if k <= n then a else b, k, 1, n)); 712a * n$ 713 714(declare(nn,integer), sum(sin(%pi * k),k,0,nn)); 7150$ 716 717sum(sin(x),x,1/2,1/2 + nn); 718'sum(sin(x),x,1/2,1/2 + nn); 719 720remove (nn, integer); 721done; 722 723(assume_pos: assume_pos_save, apply (forget, facts ()), 0); 7240; 725 726nusum ((n - 1)*(n - 1)!, n, 1, n); 727n! - 1; 728 729nusum (n!, n, 1, n); 730'sum (n!, n, 1, n); 731 732nusum ((2*m)!/(m!*(m + 1)!), m, 0, n); 733'sum ((2*m)!/(m!*(m + 1)!), m, 0, n); 734 735 736/*--- A test for sumcontract ---*/ 737 738sumcontract(sum(k,k,1,n) + sum(k^2, k, 1, n)); 739sum(k^2+k,k,1,n); 740 741/* Bug ID: 2770575 - rtestsum test 226 742 * This test simplifies correctly after revision 1.12 of sumcon.lisp. 743 */ 744sumcontract(sum(k, k, 1, n) + sum(k, k, 1, n+1)); 745n+2*sum(k,k,1,n)+1; 746 747/* Tests for bug report [ 1497706 ] sum(1/k^2,k,2,inf), simpsum; */ 748 749/* sum (1/k^2, ...) not simplified unless specifically requested */ 750(sum (1/k^2, k, 1, inf), [op (%%), args (%%)]); 751[''(nounify (sum)), [1/k^2, k, 1, inf]]; 752 753sum (1/k^2, k, 1, inf), simpsum; 754%pi^2/6; 755 756/* this is the case cited in the bug report */ 757sum (1/k^2, k, 2, inf), simpsum; 758%pi^2/6 - 1; 759 760sum (1/k^2, k, 1, 9); 7619778141/6350400; 762 763sum (1/k^2, k, 10, inf), simpsum; 764%pi^2/6 - 9778141/6350400; 765 766sum (1/k^4, k, 1, 5); 76714001361/12960000; 768 769sum (1/k^4, k, 6, inf), simpsum; 770%pi^4/90 - 14001361/12960000; 771 772sum (1/k^7, k, 1, inf), simpsum; 773zeta (7); 774 775sum (1/k^7, k, 1, 3); 776282251/279936; 777 778sum (1/k^7, k, 4, inf), simpsum; 779zeta (7) - 282251/279936; 780 781/* Tests for bug report [ 1550985 ] niceindices 782 * Test the cases mentioned in the bug report & some others for good measure 783 */ 784 785(reset (niceindicespref), 0); 7860; 787 788niceindices (sum (1/kk, kk, 1, n) - sum (1/ii, ii, 1, n)); 7890; 790 791niceindices (sum (1/k, k, 1, n) - sum(1/i, i, 1, n)); 7920; 793 794niceindices (sum (F(foo), foo, 1, inf)); 795'sum (F(i), i, 1, inf); 796 797(niceindicespref : '[foo, bar, baz, quux], 0); 7980; 799 800niceindices (sum (1/kk, kk, 1, n) - sum (1/ii, ii, 1, n)); 8010; 802 803niceindices (sum (1/k, k, 1, n) - sum(1/i, i, 1, n)); 8040; 805 806niceindices (sum (F(foo), foo, 1, inf)); 807'sum (F(foo), foo, 1, inf); 808 809niceindices (product (sum (product (sum (product (m*F(i) + cos(G(j))^k/sin(l), m, 1, n5), l, 1, n4), k, 1, n3), j, 1, n2), i, 1, n1)); 810'product ('sum ('product ('sum ('product (foo*F(foo0) + cos(G(quux))^baz/sin(bar), foo, 1, n5), bar, 1, n4), baz, 1, n3), quux, 1, n2), foo0, 1, n1); 811 812niceindices (sum (sum (foo/foo0, foo, 1, n), foo0, 1, m)); 813('sum (1/foo, foo, 1, m)) * 'sum (foo, foo, 1, n); 814 815niceindices (sum (sum (1/(foo + foo0), foo, 1, n), foo0, 1, m)); 816'sum ('sum (1/(foo + bar), foo, 1, n), bar, 1, m); 817 818(reset (niceindicespref), 0); 8190; 820 821/* Tests for bashindices, which used to not transform all indices in nested 822 * sums and products (only the outermost) 823 */ 824 825(reset (gensumnum), 0); 8260; 827 828bashindices (sum (product (sum (F (l, m, n), l, 0, m), m, 0, n), n, 0, N)); 829'sum ('product ('sum (F (j1, j2, j3), j1, 0, j2), j2, 0, j3), j3, 0, N); 830 831(reset (gensumnum), 0); 8320; 833 834/* Test for bug report [ 1552710 ] product(sum(f(i),i,1,inf),j,1,inf) => inf (wrong) */ 835 836/* Since sum(f(i), i, 1, inf) is free of j, another result 837 * such as limit(sum(f(i), i, 1, inf)^n, n, inf) would be OK, too. 838 * In any event, inf (observed in the bug report) is not OK. 839 */ 840(kill(f), block ([x : product (sum (f(i), i, 1, inf), j, 1, inf)], block ([simp : false], is (x = 'product ('sum (f(i), i, 1, inf), j, 1, inf))))); 841 842true; 843 844/* Test for bug reported by Amit Aronovitch to mailing list 2006/01/06 845 */ 846 847(kill(f), f(x) := sum (x^n/n!, n, 0, inf), f(2*x)); 848'sum (2^n*x^n/n!, n, 0, inf); 849 850 851/* Test for intosum */ 852 853intosum(2*sum(k,k,1,inf)); 854sum(2*k,k,1,inf); 855 856/* correct a bug in SUBST-IF-NOT-FREEOF (src/asum.lisp): 857 * array flag in CAR was lost before, but it's OK now 858 */ 859(kill (a, i), x : a[i], sum (x, i, 1, 3)); 860a[1] + a[2] + a[3]; 861 862/* Restore treatment of definite summations as it was before summation 863 * revisions of December 2005: simplify definite summation to explicit 864 * sum only if simpsum is true. 865 */ 866 867(kill (f), sum (f(i), i, 1, 3)); 868f(1) + f(2) + f(3); 869 870'sum (f(i), i, 1, 3); 871'sum (f(i), i, 1, 3); 872 873'sum (f(i), i, 1, 3), simpsum; 874f(1) + f(2) + f(3); 875 876/* (analogous tests for product) */ 877 878product (f(i), i, 1, 3); 879f(1)*f(2)*f(3); 880 881'product (f(i), i, 1, 3); 882'product (f(i), i, 1, 3); 883 884'product (f(i), i, 1, 3), simpproduct; 885f(1)*f(2)*f(3); 886 887/* Differentiate wrt a subscripted symbol in a summation. 888 * Should work the same as for an ordinary symbol. 889 */ 890(kill (x, a, a1), 0); 8910; 892 893diff ('sum (sin (a1 * x[i]), i, 1, n), a1); 894'sum (x[i] * cos (a1 * x[i]), i, 1, n); 895 896diff ('sum (x[i]^a1, i, 1, n), a1); 897'sum (x[i]^a1 * log(x[i]), i, 1, n); 898 899diff ('sum (sin (a[1] * x[i]), i, 1, n), a[1]); 900'sum (x[i] * cos (a[1] * x[i]), i, 1, n); 901 902diff ('sum (x[i]^a[1], i, 1, n), a[1]); 903'sum (x[i]^a[1] * log(x[i]), i, 1, n); 904 905/* Feature request 1848704: no way to convert log(product(...)) to sum(log(...)...) 906 */ 907 908(kill (u, a, x, T), 0); 9090; 910 911log (product (u[k], k, 1, n)); 912log (product (u[k], k, 1, n)); 913 914log (product (u[k], k, 1, n)), logexpand=true; 915log (product (u[k], k, 1, n)); 916 917log (product (u[k], k, 1, n)), logexpand=all;sum (log (u[k]), k, 1, n); 918 919log (product (u[k], k, 1, n)), logexpand=super; 920sum (log (u[k]), k, 1, n); 921 922log (product (a*%e^x(i), i, 1, T)); 923log (product (a*%e^x(i), i, 1, T)); 924 925log (product (a*%e^x(i), i, 1, T)), logexpand=all; 926/* Yields sum(const + ...) instead of const*T + sum(...) 927 * since sum is not known to be linear by default. 928 * Change this result if default ever changes. 929 */ 930sum (log(a) + x(i), i, 1, T); 931 932/* bug reported to mailing list 2008-04-17: 933 * simpsum causes two evaluations of summand, should be only one 934 */ 935(bar (x, y) := sum (F (i, x, y), i, 1, n), 936 ev (bar (y, x), simpsum=false)); 937'sum (F (i, y, x), i, 1, n); 938 939ev (bar (y, x), simpsum=true); 940'sum (F (i, y, x), i, 1, n); 941 942/* original problem */ 943 944(kill (all), 945 geom(x,m,n):= (x^m-x^(n+1))/(1-x), 946 B(s,sp):=-phi[1](s)*phi[1](s+sp)*phi[0](s+sp)/(1-phi[1](s+sp)*phi[0](s +sp)) 947 +(1-phi[1](s))*sum((phi[1](s)*phi[0](s))^n*( 948 geom( (phi[0](s+sp)*phi[1](s+sp))/(phi[0](s)*phi[1](s)), 1, n-1 ) 949 -phi[1](s+sp)/phi[1](s) 950 *geom((phi[0](s+sp)*phi[1](s+sp))/(phi[0](s)*phi[1](s)), 0, n-1)) 951 , n , 1, inf ), 952 A(s):=phi[1](s)*(1+phi[0](s))/(1-phi[1](s)*phi[0](s)), 953 C(s,sp):=A(s+sp)+B(s,sp)+B(sp,s), 954 0); 9550; 956 957(expr_1 : ev (C (s, sp), simpsum=false), 958 expr_2 : ev (C (s, sp), simpsum=true), 959 is (expr_1 = expr_2)); 960true; 961 962(expr_1 : ev (C (sp, s), simpsum=false), 963 expr_2 : ev (C (sp, s), simpsum=true), 964 is (expr_1 = expr_2)); 965true; 966 967is (sublis ([s=sp, sp=s], C (s, sp)) = C (s, sp)), simpsum; 968true; 969 970is (sublis ([s=sp, sp=s], C (sp, s)) = C (sp, s)), simpsum; 971true; 972 973is (C (sp, s) = C (s, sp)), simpsum; 974true; 975 976/* Tests for fbino in combin.lisp */ 977sum(binomial(2*n-k,k), k, 0, n), simpsum; 978fib(2*n+1); 979 980sum(binomial(n-k,k), k, 1, n), simpsum; 981fib(n+1)-1; 982 983(assume(n>1), 0); 9840; 985 986sum(binomial(n-k,k+1), k, 1, n-1), simpsum; 987fib(n+2)-n-1; 988 989sum(binomial(2*n,2*k), k, 0, n), simpsum; 9902^(2*n-1); 991 992sum(binomial(n,2*k+1), k, 0, n), simpsum; 9932^(n-1); 994 995sum(binomial(n,2*k), k, 0, n), simpsum; 9962^(n-1); 997 998sum(binomial(n,2*k), k, 1, n), simpsum; 9992^(n-1)-1; 1000 1001sum(binomial(n+1,k), k, 1, n), simpsum; 10022^(n+1)-2; 1003 1004sum(binomial(n,k), k, -1, n+1), simpsum; 10052^n; 1006 1007sum(binomial(t+i,t),i,0,k), simpsum; 1008binomial(t+k+1,t+1); 1009 1010/* Because of revision 1.43 of csimp2.lisp binomial(x,x) no longer simplifies 1011 * automatically to 1. The sign must be known not be negative. 1012 * Therefore, we add a fact about the sign of t. 1013 */ 1014(assume(t>0),done); 1015done; 1016sum(binomial(t+i,t),i,1,k), simpsum; 1017binomial(t+k+1,t+1)-1; 1018(forget(t>0),done); 1019done; 1020 1021sum(binomial(t+i,i),i,0,k), simpsum; 1022binomial(t+k+1,t+1); 1023 1024(forget(n>1), 0); 10250; 1026 1027/* bug in assume database causes trouble for conditional in summation 1028 * see mailing list 2009-07-29 "Fourier Series using fourie.mac" 1029 */ 1030 1031/* If ever the assume stuff is revised, this next test can go away. 1032 * It's here to ensure the underlying problem is fixed. 1033 */ 1034?dcomp (1, 1.0); 1035zero; 1036 1037(kill (all), assume (xx >= 1), 0); 10380; 1039 1040is (xx > 1); 1041unknown; 1042 1043if xx > 1 then aa else bb; 1044if xx > 1 then aa else bb; 1045 1046sum (if xx > 1 then aa else bb, xx, 1, 3); 10472*aa + bb; 1048 1049foo : 'sum (if xx > 1 then aa else bb, xx, 1, 3); 1050'sum (if xx > 1 then aa else bb, xx, 1, 3); 1051 1052ev (foo, nouns); 10532*aa + bb; 1054 1055foo : sum (if equal (xx, 1) then aa else bb, xx, 1, nn); 1056'sum (if equal (xx, 1) then aa else bb, xx, 1, nn); 1057 1058ev (foo, nouns, nn=3); 1059aa + 2*bb; 1060 1061(reset(assume_pos),1); 10621; 1063