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