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| from sympy.concrete.summations import Sum | |
| from sympy.core.add import Add | |
| from sympy.core.numbers import (I, Rational, oo, pi) | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import (Symbol, symbols) | |
| from sympy.functions.combinatorial.factorials import (binomial, factorial, subfactorial) | |
| from sympy.functions.combinatorial.numbers import (fibonacci, harmonic) | |
| from sympy.functions.elementary.exponential import (exp, log) | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import (cos, sin) | |
| from sympy.functions.special.gamma_functions import gamma | |
| from sympy.series.limitseq import limit_seq | |
| from sympy.series.limitseq import difference_delta as dd | |
| from sympy.testing.pytest import raises, XFAIL | |
| from sympy.calculus.accumulationbounds import AccumulationBounds | |
| n, m, k = symbols('n m k', integer=True) | |
| def test_difference_delta(): | |
| e = n*(n + 1) | |
| e2 = e * k | |
| assert dd(e) == 2*n + 2 | |
| assert dd(e2, n, 2) == k*(4*n + 6) | |
| raises(ValueError, lambda: dd(e2)) | |
| raises(ValueError, lambda: dd(e2, n, oo)) | |
| def test_difference_delta__Sum(): | |
| e = Sum(1/k, (k, 1, n)) | |
| assert dd(e, n) == 1/(n + 1) | |
| assert dd(e, n, 5) == Add(*[1/(i + n + 1) for i in range(5)]) | |
| e = Sum(1/k, (k, 1, 3*n)) | |
| assert dd(e, n) == Add(*[1/(i + 3*n + 1) for i in range(3)]) | |
| e = n * Sum(1/k, (k, 1, n)) | |
| assert dd(e, n) == 1 + Sum(1/k, (k, 1, n)) | |
| e = Sum(1/k, (k, 1, n), (m, 1, n)) | |
| assert dd(e, n) == harmonic(n) | |
| def test_difference_delta__Add(): | |
| e = n + n*(n + 1) | |
| assert dd(e, n) == 2*n + 3 | |
| assert dd(e, n, 2) == 4*n + 8 | |
| e = n + Sum(1/k, (k, 1, n)) | |
| assert dd(e, n) == 1 + 1/(n + 1) | |
| assert dd(e, n, 5) == 5 + Add(*[1/(i + n + 1) for i in range(5)]) | |
| def test_difference_delta__Pow(): | |
| e = 4**n | |
| assert dd(e, n) == 3*4**n | |
| assert dd(e, n, 2) == 15*4**n | |
| e = 4**(2*n) | |
| assert dd(e, n) == 15*4**(2*n) | |
| assert dd(e, n, 2) == 255*4**(2*n) | |
| e = n**4 | |
| assert dd(e, n) == (n + 1)**4 - n**4 | |
| e = n**n | |
| assert dd(e, n) == (n + 1)**(n + 1) - n**n | |
| def test_limit_seq(): | |
| e = binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)) | |
| assert limit_seq(e) == S(3) / 4 | |
| assert limit_seq(e, m) == e | |
| e = (5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5) | |
| assert limit_seq(e, n) == S(5) / 3 | |
| e = (harmonic(n) * Sum(harmonic(k), (k, 1, n))) / (n * harmonic(2*n)**2) | |
| assert limit_seq(e, n) == 1 | |
| e = Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n) | |
| assert limit_seq(e, n) == 4 | |
| e = (Sum(binomial(3*k, k) * binomial(5*k, k), (k, 1, n)) / | |
| (binomial(3*n, n) * binomial(5*n, n))) | |
| assert limit_seq(e, n) == S(84375) / 83351 | |
| e = Sum(harmonic(k)**2/k, (k, 1, 2*n)) / harmonic(n)**3 | |
| assert limit_seq(e, n) == S.One / 3 | |
| raises(ValueError, lambda: limit_seq(e * m)) | |
| def test_alternating_sign(): | |
| assert limit_seq((-1)**n/n**2, n) == 0 | |
| assert limit_seq((-2)**(n+1)/(n + 3**n), n) == 0 | |
| assert limit_seq((2*n + (-1)**n)/(n + 1), n) == 2 | |
| assert limit_seq(sin(pi*n), n) == 0 | |
| assert limit_seq(cos(2*pi*n), n) == 1 | |
| assert limit_seq((S.NegativeOne/5)**n, n) == 0 | |
| assert limit_seq((Rational(-1, 5))**n, n) == 0 | |
| assert limit_seq((I/3)**n, n) == 0 | |
| assert limit_seq(sqrt(n)*(I/2)**n, n) == 0 | |
| assert limit_seq(n**7*(I/3)**n, n) == 0 | |
| assert limit_seq(n/(n + 1) + (I/2)**n, n) == 1 | |
| def test_accum_bounds(): | |
| assert limit_seq((-1)**n, n) == AccumulationBounds(-1, 1) | |
| assert limit_seq(cos(pi*n), n) == AccumulationBounds(-1, 1) | |
| assert limit_seq(sin(pi*n/2)**2, n) == AccumulationBounds(0, 1) | |
| assert limit_seq(2*(-3)**n/(n + 3**n), n) == AccumulationBounds(-2, 2) | |
| assert limit_seq(3*n/(n + 1) + 2*(-1)**n, n) == AccumulationBounds(1, 5) | |
| def test_limitseq_sum(): | |
| from sympy.abc import x, y, z | |
| assert limit_seq(Sum(1/x, (x, 1, y)) - log(y), y) == S.EulerGamma | |
| assert limit_seq(Sum(1/x, (x, 1, y)) - 1/y, y) is S.Infinity | |
| assert (limit_seq(binomial(2*x, x) / Sum(binomial(2*y, y), (y, 1, x)), x) == | |
| S(3) / 4) | |
| assert (limit_seq(Sum(y**2 * Sum(2**z/z, (z, 1, y)), (y, 1, x)) / | |
| (2**x*x), x) == 4) | |
| def test_issue_9308(): | |
| assert limit_seq(subfactorial(n)/factorial(n), n) == exp(-1) | |
| def test_issue_10382(): | |
| n = Symbol('n', integer=True) | |
| assert limit_seq(fibonacci(n+1)/fibonacci(n), n).together() == S.GoldenRatio | |
| def test_issue_11672(): | |
| assert limit_seq(Rational(-1, 2)**n, n) == 0 | |
| def test_issue_14196(): | |
| k, n = symbols('k, n', positive=True) | |
| m = Symbol('m') | |
| assert limit_seq(Sum(m**k, (m, 1, n)).doit()/(n**(k + 1)), n) == 1/(k + 1) | |
| def test_issue_16735(): | |
| assert limit_seq(5**n/factorial(n), n) == 0 | |
| def test_issue_19868(): | |
| assert limit_seq(1/gamma(n + S.One/2), n) == 0 | |
| def test_limit_seq_fail(): | |
| # improve Summation algorithm or add ad-hoc criteria | |
| e = (harmonic(n)**3 * Sum(1/harmonic(k), (k, 1, n)) / | |
| (n * Sum(harmonic(k)/k, (k, 1, n)))) | |
| assert limit_seq(e, n) == 2 | |
| # No unique dominant term | |
| e = (Sum(2**k * binomial(2*k, k) / k**2, (k, 1, n)) / | |
| (Sum(2**k/k*2, (k, 1, n)) * Sum(binomial(2*k, k), (k, 1, n)))) | |
| assert limit_seq(e, n) == S(3) / 7 | |
| # Simplifications of summations needs to be improved. | |
| e = n**3*Sum(2**k/k**2, (k, 1, n))**2 / (2**n * Sum(2**k/k, (k, 1, n))) | |
| assert limit_seq(e, n) == 2 | |
| e = (harmonic(n) * Sum(2**k/k, (k, 1, n)) / | |
| (n * Sum(2**k*harmonic(k)/k**2, (k, 1, n)))) | |
| assert limit_seq(e, n) == 1 | |
| e = (Sum(2**k*factorial(k) / k**2, (k, 1, 2*n)) / | |
| (Sum(4**k/k**2, (k, 1, n)) * Sum(factorial(k), (k, 1, 2*n)))) | |
| assert limit_seq(e, n) == S(3) / 16 | |